# Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where $A_n=[a_{ij}(n)],A=[a_{ij}]$. Let $\rho(A_n)=\{\lambda_1(n),\cdots,\ \lambda_m(n)\}$ be the eigenvalues of $A_n$ and similarly, $\rho(A)=\{\lambda_1,\cdots,\ \lambda_m\}$ be the eigenvalues of $A$, arranged in, say, increasing order. Here are my questions

1)Can I write $\lambda_k(n)\to \lambda_k,\ 1\le k\le m$?

2)If I define (with a slight abuse of standard notation) $\delta_s(n),\ 1\le s\le m$ as the maximum eigenvalue of any $s\times s$ submatrix of $A_n$ and if $\delta_s$ be the corresponding quantity for $A$, then can I say that $$\lim_{n\to \infty}\delta_s(n)=\delta_s$$ ?

Intuitively it seems to me that the answers are positive since the eigenvalues of a matrix are continuous functions of the elements of the matrix and $\delta_s$ is just the maximum of some eigenvalues of submatrices.

However, I am not sure if this argument is sound enough. Maybe this is a very trivial issue for people here, but I would really appreciate if someone can kindly provide some explanation regarding the correct answer. Thanks in advance.

• You need to be careful here; there's the possibility that the eigenvalues may "switch". For example, we could have $\lambda_1(n) \to \lambda_2$ and $\lambda_2(n) \to \lambda_1$ – Omnomnomnom Jul 2 '15 at 11:11
• Outside of that technicality, your answer is indeed correct and sufficient. – Omnomnomnom Jul 2 '15 at 11:13
• If we have symmetric matrices (as you do now), the issue of "switching eigenvalues" isn't a problem for this question. – Omnomnomnom Jul 2 '15 at 11:14
• @Omnomnomnom, is the argument I provided correct? Also, can you kindly comment on the second question? – Samrat Mukhopadhyay Jul 2 '15 at 11:14
• Yes, that's what I meant; your argument that the eigenvalues depend continuously on the matrix is enough to prove both 1) and 2). In order to get 2) quickly, we could note that the function $$(x_1,\dots,x_n) \mapsto \max \{x_1,\dots,x_n\}$$ is continuous over $\Bbb R^n$. – Omnomnomnom Jul 2 '15 at 11:17

Let $\chi_n$ be the characteristic polyomial of $A_n$ and $\chi=a\prod(X-\lambda_i)$ be that of $A$, then $\chi_n\mathop{\to}_{n\rightarrow\infty}\chi$.

If a sequence of polynomials converges to another, then by continuity the roots of the polynomials in the sequence must converge to the roots of the limit polynomial (with the same multiplicity). Hence the eigenvalues of the sequence of matrices indeed convergence to the eigenvalues of $A$.

Likewise, since any submatrix of $A_k$ converges to the corresponding submatrix of $A$, that is sufficient to say that $\delta_s(n)\to\delta_s$.

• @orangeskid Thanks for the feedback, I added an explanatory line. – Hippalectryon Jul 2 '15 at 11:25
• @Hippalectryon: It's tempting to pass to the characteristic polynomial and, since that varies continuously, use that the roots also do. That is correct but I would give it more thought. It's the problem with approaching a polynomial with multiple roots. Otherwise, the implicit function theorem works OK. – Orest Bucicovschi Jul 2 '15 at 11:30

The answer of @Hippalectryon is morally correct. However, there is this problem of approaching a polynomial with multiple roots, somehow delicate. $\tiny{\text{Can be proved with the argument principle from complex analysis.}}$ We'll work directly with the eigenvalues, bypassing the characteristic polynomial.

Use this standard observation: if two symmetric matrices are ordered $B\prec C$ then their vectors of eigenvalues ( ordered increasingly) also satisfy $\lambda(B) \prec \lambda(C)$ ( that is $\lambda_i(B) \le \lambda_i(C)$ for all $i$).

Now, if if $A_n \to A$ then for any $\epsilon >0$ we have $A- \epsilon I \prec A_n \prec A + \epsilon I$ for $n \ge n_{\epsilon}$, and so $$\lambda_i(A) - \epsilon \le \lambda_i(A_n) \le \lambda_i(A) + \epsilon$$

$\tiny{\text{(the question for the max for submatrices is simple:$\lim$and$\max$commute ).}}$

$\bf{Added:}$ I am giving this " continuity of the roots" question more thought. Of course, @Hippalectryon is right. But how do we truly convince ourselves that the roots of $P_n$ approach the roots of $P$? First, a hands-on approach: The roots of $P_n$ will all lie in a bounded region since the coefficients of $P_n$ are all bounded. If the roots of $P_n$ did not approach those of $P$ then we would find, by compactness, a subsequence whose roots approach some other $n$-uple. But that would mean in the limit that $P$ would decompose using that $n$-uple, that is, it would have two distinct decompositions, contradiction. The high-brow explanation is that the map roots $\mapsto$ polynomials from $\mathbb{C}^n$ to $\mathbb{C}^n$ is continuous and proper and induces a bijective map $\mathbb{C}^n/S_n \to\mathbb{C}^n$, which is bijective, continuous and closed, hence a homeomorphism. $\tiny{\text{(surjectivity is equivalent to the fundamental theorem of algebra)}}$

• By $A\prec B$ do you mean that the elements of $A$ are smaller than the elements of $B$, individually? – Samrat Mukhopadhyay Jul 4 '15 at 5:41
• @Samrat Mukhopadhyay: I mean $B-A$ positive definite. – Orest Bucicovschi Jul 4 '15 at 8:30
• Oh, ok, got it. – Samrat Mukhopadhyay Jul 4 '15 at 11:37


# Why would the author ask if I used the Associative Law to prove + is not equiv. to *?

I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows:

Problem 1: Show that + and * are necessarily different operations. That is, for any system (F, +, *) satisfying Axioms I, II, and III, it cannot happen that x + y = x * y for all x, y. Hint: You do not know there are any numbers other than 0 and 1, so that your argument should probably involve only these numbers. Did you use Axiom II? If not, state explicitly the stronger result that you actually proved.

In this book, Axiom I is commutativity of + and *, Axiom II is associativity of + and *, and Axiom III is existence of identities (x+0=x, x*1=x, 0 does not equal 1).

My question: Simply why would the author specifically ask the reader if he/she used Axiom II (associativity) and what exactly do they mean by "If not, state explicitly the stronger result you actually proved"? Why not not include those last two sentences?

FWIW, here is my solution:

To prove: Restated:

And I justified 5 by citing Axiom III since Axiom III includes the statement that 0 does not equal 1.

-
Hint: if you didn't use Axiom II, what can you now say about systems where Axioms I And III hold but II possibly does not? – Steven Stadnicki Jun 5 '12 at 16:13
More succinctly: $\ 1 = 1 + 0 = 1 * 0 = 0.\$ Note that the proof requires only one of the operations to be commutative, so it will work for noncommutative rings too, i.e. where multiplication is not necessarily commutative. It wouldn't require comutativity at all if Axiom II was $\rm\:1*x = x\:$ vs. $\rm\:x*1 = x.$ Said simply, the hypothesis $\:1\ne 0\:$ forces addition $\ne$ multiplication, because these operations have the value $1,0$ resp. at the same point, viz. $(1,0)$ – Bill Dubuque Jun 5 '12 at 16:16
Since I didn't use Axiom II (associativity), my answer to the request "State explicitly the stronger result that you actually proved" is "The non-equivalence of + and * is not dependent on associativity holding in the system F." Is that a good answer to the meaning of the very last sentence of the problem? – mring Jun 5 '12 at 16:42
@Pete Close, though I would make it clear that it does still require Axioms I and III, e.g. (though this is awkward) '+ and * are non-equivalent in each system with commutativity and existence of identities, regardless of associativity'. – Steven Stadnicki Jun 5 '12 at 17:04

The basic idea is as follows. From the neutral Axioms III, and commutativity of addition we have

$$\begin{eqnarray}\rm x = 0 + x &\rm \\ \rm y *\: 1 &=\rm y \end{eqnarray}$$

If $\ +\, =\, *\$ then aligned terms are unified for $\rm\:y = 0,\ x = 1,\:$ yielding

$$\rm\ 1 = 0 + 1 = 0 * 1 = 0$$

contra hypothesis $\rm\:1 \ne 0.\:$ Thus $\rm\: +\: \ne\: *\:$ because they take different values at the point $\rm\:(0,1)$.

Note that the proof does not use associativity, and doesn't use commutativity if you state the neutral axioms as above. In any case, only one of the commutative axioms is needed, so that the neutral axioms can be ordered so the above unification is possible. In particular, the inference works in noncomutative rings, i.e. rings where multiplication is not necessarily commutative. Further, because the proof did not use associativity, it will also work in nonassociative rings.

Note $\$ This method of deriving consequences by unifying terms in identities is a basic method in equational reasoning (term rewriting), e.g. google Knuth-Bendix or Grobner basis algorithms.

-

The author probably wants to add that it is not necessary to use Axiom II in order to prove that the laws must be different.

If you then forget about axiom II, and if you suppose that + and * are the same, your axioms become :

axiom 1 : $x+y = y+x$
axiom 3a : $x+0 = x$
axiom 3b : $x+1 = x$
axiom 3c : $0 \neq 1$.

In order to get a contradiction you must use axiom 3c, and the only way to use it is to show that axioms 1,3a,3b implies that $0 = 1$.

So you really are investigating commutative laws with two identity elements, and in fact, you necessarily have to prove that the identity element of a commutative law must be unique :

If 0 and 1 are identity elements of +, then 0 = 0+1 = 1+0 = 1, so they are the same. There, we proved that if a commutative law has an identity element, it is unique.

-

Proven without Associativity:

Given the two different operations, $x+y\neq x*y$ cannot hold $\forall x,y$ even when the order of evaluation matters (the case where Axiom II is false, e.g. subtraction and division).

-

As far as I understand this the author also wants to teach how to understand/write a proof in general. If you are given a proof of a statement a good place to start understanding it, is figuring out where exactly every single assumption was used. The second question is which assumptions didn't we use and whether we actually proved a stronger result.

This is a very useful lesson for your future as a mathematician! The first question is probably more important when it comes to understanding a specific proof, while the second question may help you linking the statement with other similar statements. Obviously in your rather elementary exercise it might not reveal its full importance but as soon as things get more complicaated it is a good idea to keep these two questions in mind.

-


# How to find a [2D] vector which is perpendicular to a line and points to a specific half-plane?

Given three points $$A$$, $$B$$ and $$C$$, that are not colinear, I want to find the vector $$v$$ which is perpendicular to the line $$L$$ that passes from $$A$$ and $$B$$ AND points to the half-plane created by $$L$$ that $$C$$ lies on.

I know that there are two orthogonal vectors to $$T$$: $$\vec{v_1} = [t_y, -t_x]$$ and $$\vec{v_2} = [-t_y, t_x]$$. I can find equation of the line, like $$L: y=ax+c$$, and solve it for both $$C_x$$ and $$t_y+A_x$$. I will return $$\vec{v_1}$$ if results had same sign, and $$\vec{v_2}$$ otherwise. And, of course, I have to take care of special case where $$L: y=c$$. Searching for an algebraic solution, I found this question: Find closest vector to A which is perpendicular to B. I tried to simplify the accepted answer for 2D, which resulted in:

\begin{aligned} s&=u\times t = (u_xt_y-u_yt_x)\vec{k} = s_z\vec{k}\\ v&=t\times s = s_z(t_y\vec{i}-t_x\vec{j}) = s_z\vec{v_1} \end{aligned}

That can be written as following, if the magnitude of the resulting vector is unimportant: $$v = \begin{cases} \vec{v_1}, & \text{if s_z > 0} \\[2ex] \vec{v_2}, & \text{if s_z<0} \\[2ex] \vec{0}, & \text{if s_z =0} \end{cases}$$

But I'm not sure about the followings:

1. Is the linked question really relevant to my problem? It says the closest vector that ..., which I don't even know what it means.
2. Did I do the math right?
3. Is there an even faster (less logical and floating point operations when implementing it) way to choose one of $$\vec{v_1}$$ and $$\vec{v_2}$$? I really don't care about the magnitude of the resulting vector.

## 2 Answers

The vectors $$v_1$$ and $$v_2$$ are indeed both nonzero and perpendicular to $$L$$, and also point in opposite directions, so one of them has to be a "good" one.

If you compute $$u = C - A$$, then you can compute $$h = u \cdot v_1,$$ the dot product of $$u$$ and $$v_1$$.

If this turns out positive, then $$u$$ and $$v_1$$ point into the same halfplane, and your answer is $$v_1$$; if it's negative, your answer is $$v_2$$. If it's $$0$$, then $$C$$ is actually collinear with $$A$$ and $$B$$, which is kind of a free sanity-check that your inputs were valid. [All this is a rehash of what you wrote in your question as the first displayed equation.]

An alternative approach is to compute $$s = u \times t$$ (in 3-space), which seems bad because it's a cross-product, which looks like 2 multiplies and a subtraction for each term...but you only need to compute the "z" term because the other two are always zero. Then you compute $$v = t \times s$$, and this vector will point in the right direction. You needn't compute the $$z$$-component (it'll be zero), so you end up doing a total of $$6$$ multiplies and $$3$$ subtracts, plus the 6 subtractions to compute $$u$$ and $$t$$ in the first place.

Is this a winning algorithm? I haven't counted the operations in the other one. But it has one advantage: there's no branching, which can be helpful on some highly-parallel architectures (or at least this used to be the case).

It also has a downside: if $$C$$ is very close to the line $$AB$$, the magnitude of the resulting vector gets very small. You have to decide whether that matters to you. You said not, but in practice, I've often found such things annoying.

Here's something like an algorithm, written in very explicit Matlab

function v = findVec(A, B, C)
% Given points A,B,C in the xy-plane, C not on the line AB, find
% a vector v in the xy-plane that is perpendicular to AB, and points
% into the halfplane containing C

u = [C(1) - A(1), C(2) - A(2)];
t = [B(1) - A(1), B(2) - A(2)];

s = [0, 0, u(1)*t(2) - u(2)* t(1)]; % first cross product
v = [t(2)*s(3), -t(1) * s(3)];

and here's the more idiomatic, minimal operations, version:

function v = findVec(A, B, C)
% Given points A,B,C in the xy-plane, C not on the line AB, find
% a vector v in the xy-plane that is perpendicular to AB, and points
% into the halfplane containing C

trot = [B(2) - A(2), A(1) - B(1)]; % the t vector, rotated 90 degrees
% because that's what I'll need in a minute.

sz = (C(1) - A(1)) * trot(1) + (C(2) - A(2)) * trot(2);
v = sz * trot;

• Thank you very much John. I actually made a terrible mistake in my calculations and probably misled you. Because in fact the equation I put was obtained from your second method! I edited my question. But it was very beautiful that in the end both methods came to the same result, since u.v1=sz=(ux*ty-uy*tx). However, I prefer to look at it as the result of the dot product, thanks again for bringing it up. Oct 6 '20 at 6:09
• And there was an H after your first equation. Did you mean u or I'm missing something? And I believe that's s(3) in your first code sample. Oct 6 '20 at 6:22
• Fixed both --- sorry about the glitches. Oct 6 '20 at 11:10

The component of $$\vec{u}$$ that is parallel to $$\vec{t}$$ is

$$\vec{u}_\parallel = \frac{ \vec{t} \cdot \vec{u} }{ \| \vec{t} \|^2 }\, \vec{t}$$

So subtract it form $$\vec{u}$$ to get $$\vec{v}$$

$$\vec{v} = \vec{u} - \frac{ \vec{t} \cdot \vec{u} }{ \| \vec{t} \|^2 } \, \vec{t}$$

Proof

$$\vec{t} \cdot \vec{v} = \vec{t} \cdot \vec{u} - \frac{ \vec{t} \cdot \vec{u} }{ \| \vec{t} \|^2 } \, (\vec{t} \cdot \vec{t}) = \vec{t} \cdot \vec{u} -\vec{t} \cdot \vec{u} = \vec{0}$$

• Thanks John. That is aد interesting solution. But the calculations needed to get the answer from this method are huge. That norm in the denominator really terrifies me :) Oct 6 '20 at 6:37


# How to code Adams-Bashforth method

Posted 1 year ago
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## Original Question

There used to be a question about a manual implementation of the Adams-Bashforth method of numerically integrating an ODE. The code looked like it had be translate from C. I started to write up a more functional-programming to the problem. Well, why not share what I had done? Presumably the purpose of the writing the solver is educational. I will keep the design simple. For a more robust approach, I would recommend working within the NDSolve framework.

The original code looked something like this:

MyAdamBashforth[F_, h_, t_, y_][{t0_, y0_}] :=
Module[{j, p, m, a, F0, F1, F2},
m = 8;
a = 0;
t[0] = t0;
y[0] = y0;
For[j = 0, j <= 1, j++,
t[j + 1] = t[j] + h;
y[j + 1] = y[j] + h*F[t[j], y[j]];
];
F0 = F[t[0], y[0]];
F1 = F[t[1], y[1]];
F2 = F[t[2], y[2]];
For[j = 2, j <= m, j++,
p = y[j] + h/12 (5 F0 - 16 F1 + 23 F2);
t[j + 1] = t[j] + h;
y[j + 1] = p;
F0 = F1;
F1 = F2;
F2 = F[t[j + 1], y[j + 1]];
];
]

The solver was used in a way like this:

Clear[F, y, t, h];
F[t_, y_] := t + y;
MyAdamBashforth[F, 0.5, t, y][{0., 1.}]
MyData3 = Table[{t[j], y[j]}, {j, 0, 8}]

(*  {{0., 1.}, {0.5, 1.5}, {1., 2.5}, {1.5, 4.72917}, {2., 8.78212},
{2.5, 15.6914}, {3., 27.2344}, {3.5, 46.3278}, {4., 77.713}}  *)

## Design choices

In general, the (discrete) solution of an ODE $x'(t) = \text{RHS}(t, x(t))$ over $a \le t \le b$ consists of step data

$$t_j, x_j, x'_j,\dots \quad \text{for}\ j = 0, 1, \dots, n\,,$$

where $t_0=a$, $t_n=b$, and the number of derivative values $x'_j, x''_j,\dots$ store for each step, if any, may depend on the method. For a first or higher order ODE, $x'_j$ is calculated anyway and may be used for cubic Hermite interpolation between the steps. In a multistep method, the $j+1$ step is computed from some number of the previous steps; single-step methods use only step $j$. The Adams-Bashforth (AB) method is a linear $s$-step method that uses only the values of $x'$ from the previous $s$ steps.

Input: Representation of the problem

Following the original question, the ODE will be specified by a function rhs that represents the first-order equation x'[t] == rhs[t, x[t]]. The initial condition (IC) is simply two numbers t0, x0, representing x[t0] == x0. Also we will require a final time t1, a step size h, and the step order s.

Output data structure

The output will be a list of the step data. The data for a step consists of $t, x, p=x'$. Neither the time $t$ nor the derivative $p$ are strictly necessary to keep, but they can be convenient to keep for the sake of interpolation. If efficiency is a premium, then it might make sense to try to generate a packed array {{t, x, p},...}. If the data is to be passed to Interpolation[], then it has to be put in the form {{{t}, x, p},...}, which cannot be packed. Since this is an educational exercise, I'll choose the latter.

The solver

The basic idea is that the solution data could be generated by Nest[] or NestList[] with a call like

Nest[abStep[rhs, h, s], {IC}, nsteps]

where abStep[rhs, h, s][stepdata] computes next step using the s-step method, IC is the initial condition, and nsteps is the number of steps to traverse the interval of integration. Whether to use Nest[] or NestList[] depends on choices involving the iterator abStep.

The iterator

One constraint with the choice of nesting, is that the output of abStep has to be valid input for it. The function abStep has two phases:

• initialization steps: until s steps are generated, another method is used to compute the next step;
• s-step steps: once s are generate, use the AB s-step method to compute the next step.

Since AB is a multistep method, abStep needs access to previous steps. It only needs the last s steps, but I chose to pass it all steps. This allows the constraints of the problem by coding abStep to return the list of past steps with the next step appended to it. With that design, the solver calls the iterator with Nest[] and not NestList[].

Initialization

The original question uses Euler steps for the initialization of the s-step phase. It was easier to code abStep to use AB with a step order equal to the number of steps so far (up to s). This should be more accurate but a little slower, two considerations that are of minor importance in this exercise. However one wants to compute the initial steps, it is not hard to write a function abStep to do it, since all steps are passed to it, and it is easy to select a method based on how many steps are passed.

## Implementation

ClearAll[abCoeffs];            (* Adams-Bashforth coefficients *)
mem : abCoeffs[s_] :=  mem =   (* for mem, see note on memoization below *)
Table[(-1)^(s - j)/((j - 1)! (s - j)!) * Integrate[Product[If[i == s - j, 1, u + i], {i, 0, s - 1}], {u, 0, 1}], {j, s}];

Clear[abStep];                         (* Adams-Bashforth step of order at most s *)
abStep[rhs_, h_, s_][past_] :=         (* past = previous steps = {{{t}, x, p}..} *)
With[{steps = Min[s, Length@past]},  (* step order cannot exceed number of steps taken *)
With[{  (* next step *)
t0 = past[[-1, 1, 1]] + h,
x0 = past[[-1, 2]] + h*abCoeffs[steps].past[[-steps ;;, 3]] (* Dot product gives the AB linear comb. of coefficients & steps *)
},
With[{p0 = rhs[t0, x0]},           (* store the derivative value; used in following step, too *)
Append[past, {{t0}, x0, p0}]      (* add step to past ones *)
]]];

Clear[abSolve];   (* Adams-Bashforth solver x'[t] == rhs[t,x[t]] *)
abSolve[rhs_, x0_, {t0_, t1_, h_}, s_] :=
Nest[
abStep[rhs, h, s],
{{{t0}, x0, rhs[t0, x0]}},  (* initial value *)
Floor[(t1 - t0)/h]          (* truncates interval like Range[t0, t1, h]; Ceiling[] is an alternative *)
];

## Examples

One can check the formula with the original question as follows:

Clear[F];
abStep[F, h][{{t}, y, F2, F1, F0}]
(*  {{h + t}, ((5 F0)/12 - (4 F1)/3 + (23 F2)/12) h + y, F[h + t, ((5 F0)/12 - (4 F1)/3 + (23 F2)/12) h + y], F2, F1}  *)

Here is the original numerical example, with 2-step and 3-step solutions:

ClearAll[F];
F[t_, y_] := t + y;
sol2 = Interpolation@abSolve[F, 1., {0., 4., 0.5}, 2];
sol3 = Interpolation@abSolve[F, 1., {0., 4., 0.5}, 3];

sol0 = y /. First@DSolve[{y'[t] == F[t, y[t]], y[0] == 1}, y, t];
Show[
Plot[{sol0[t]}, {t, 0, 4}, PlotLegends -> {"Exact"}],
Plot[{Undefined, sol3[t]}, {t, 0, 4}, Mesh -> sol3["Coordinates"],
PlotLegends -> {None, "3-step"}],
Plot[{Undefined, Undefined, sol2[t]}, {t, 0, 4},
Mesh -> sol2["Coordinates"], PlotLegends -> {None, None, "2-step"}]
]

The data itself:

abSolve[F, 1., {0., 4., 0.5}, 3]
(*
{{{0.}, 1., 1.},           {{0.5}, 1.5, 2.},          {{1.}, 2.75, 3.75},       {{1.5}, 5.21875, 6.71875},
{{2.}, 9.57422, 11.5742}, {{2.5}, 16.9683, 19.4683}, {{3.}, 29.3089, 32.3089}, {{3.5}, 49.7041, 53.2041},
{{4.}, 83.208, 87.208}}
*)

## Appendix: Memoization

Note: Memoization. "Memoization" or "cacheing" is a technique to store a computed function value so that on subsequent calls, the value need not be recomputed, only recalled from memory. For function values that are used frequently, this can save considerable time. One has to weigh this savings against greater memory usage. I learned the form mem : f[x_,...] := mem = ... on StackExchange. The mem : ... is a named the pattern and it matches the actual function call.

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Posted 1 year ago
 Thank you very much Michael Rogers.It was a question by me. :)Best wishes Zharou Fisher
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the basic outline of the more complex figure is easily arrived at, then details can be added as necessary, but the figure is already recognizable for what it is. This video looks at graphing simple absolute value functions by hand. See y = 2|x| - 3.There's |x|.So draw y = 2x - 3at x > 0. This vertex is also the lowest point on the graph. Definition of the Absolute Value. Yes, they always intersect the vertical axis. 0. Save. He posts almost all his work as that user. 0. example. Graphing absolute value function given below. If we plot these points on the graph sheet, we will get a graph as given below. Have you ever tried to draw a picture of a rabbit, or cat, or animal? The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. Next, a student will read the objective, "SWBAT graph absolute value functions on a coordinate plane". This leads to two different equations we can solve independently. Share practice link. There is an easy procedure you can follow to graph absolute value equations: Plot the value of x such that the expression inside the absolute value bars is 0. If we couldn’t observe the stretch of the function from the graphs, could we. These can be achieved by first starting with the parent absolute value function, then shifting the graph according to function transformations, flip graph if necessary and even may have to compress or decompress the graph. Learn vocabulary, terms, and more with flashcards, games, and other study tools. (c) The absolute value function intersects the horizontal axis at two points. Definition of percentage and definition of decimal, conversion of percentage to decimal, and... Robert Langlands: Celebrating the Mathematician Who Reinvented Math! Calculate |f(x)| and f(|x|): Quadratic Equation Note: Adding a positive number after the x inside the parentheses shifts the graph left, adding a negative (or subtracting) shifts the graph right. 0% average accuracy. DRAFT. a) (-9, 0)  \quad\quad\quad  b) (9, 0)  \quad\quad\quad  c) (0, -9) Play. Describe the Transformations using the correct terminology. Each coordinate pair represents the vertex of the graph of an absolute value function. Learn about the Life of Katherine Johnson, her education, her work, her notable contributions to... Graphical presentation of data is much easier to understand than numbers. Mathematics. 10th grade . The following steps will be useful to graph absolute value functions. 4 minutes ago. For example, |-7|=7 , |1|=1 ,and |-4|=4 ; the pattern is quite clear: the absolute value of a negative number is positive.In this section we consider the graph of f(x) and discuss how to graph |f(x)| and f(|x|) .. No, they do not always intersect the horizontal axis. Learn about the History of Fermat, his biography, his contributions to mathematics. This is an important function transformation. Each coordinate pair represents the vertex of the graph of an absolute value function. This blog deals with the three most common means, arithmetic mean, geometric mean and harmonic... How to convert units of Length, Area and Volume? Different types, Formulae, and Properties. These steps should be kept in mind in graphing absolute value function. Vertical and Horizontal Function Transformations. Because of this "minus", the positive values provided by the absolute-value bars will all be switched to negative values. Cuemath, a student-friendly mathematics and coding platform, conducts regular Online Live Classes for academics and skill-development, and their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills. Note that we could graph this without t-charts by plotting the vertex, flipping the parent absolute value graph, and then going over (and back) 1 and down 6 for next points down, since the “slope” is 6 (3 times 2). Complete Guide: How to multiply two numbers using Abacus? Note: Adding a positive number after the x outside the parentheses shifts the graph up, adding a negative (or subtracting) shifts the graph down. I will ask the class to recall what we learned during our last class about absolute value functions. Learn about Operations and Algebraic Thinking for grade 3. Save. We have already understood in detail about the absolute function in the blog about the absolute value function. 10 Type your answer below and then sketch a "good enough" graph of h(x) -5 -10 -5 0 5 10 Submit … This blog explains how to solve geometry proofs and also provides a list of geometry proofs. When we look at the above graph, clearly the vertex is (0, 0), Step 2: Write the given absolute value function as $$y - k = |x - h|$$. Hence, graphing absolute value functions is an important topic which we will reduce to a step by step easy process. Write the given absolute value function in the form : To get the vertex, equate (x - 1) and (y + 2) to zero. The following steps will be useful in graphing absolute value functions. This blog deals with various shapes in real life. As well as this each function has a vertex, the point where the come together. Learn about the Conversion of Units of Speed, Acceleration, and Time. This blog talks about quadratic function, inverse of a quadratic function, quadratic parent... Euclidean Geometry : History, Axioms and Postulates. Graphing absolute value functions. Home Page Describe the Transformations using the correct terminology. Learn about real-life applications of fractions. by m_13539377_87591. Using pizza to solve math? There's |x|. The constant of the function will determine whether the graph will translate up or down. Below is the graphing of absolute value equationThe graph of $$y= |x|$$ has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. Highlights common student mistakes and points out how to … The first half of the video utilizes function translations of a base function to create the sketch of the … Graphing Absolute Value Functions. … The given absolute value function is in the form : To get the vertex, equate (x - 1) and y to zero. Absolute Value Graphs & Transformations • … World cup math. To graph an absolute value function, choose several values of x and find some ordered pairs. The 3 Means: Arithmetic Mean, Geometric Mean, Harmonic Mean. This is a step by step tutorial on how to graph functions with absolute value. Example 3. This page explains how to keep track of the "V" in the absolute-value graph, and how to draw the correct shape for the quadratic equation's parabola. Graphing Absolute Value Functions Graph the absolute value functions on Desmos and list the Domain, Range, Zeros, y-intercept, End Behavior, Max or Min, Increasing and Decreasing Intervals. g (x) = f (x) + k. When k > 0, the graph of g (x) translated k units up. Graphing Absolute Value of Functions DRAFT. Practice: Graph absolute value functions. (see graph below). Absolute value graphs review. Our strategy in the next example is to make liberal use of Definition 2.4 along with what we know about graphing linear functions (from Section 2.1 ) and piecewise-defined functions (from Section 1.4). This section will focus on two particular types of transformations: vertical shifts and horizontal shifts. under the reflection in the y-axis. If we are unable to determine the stretch based on the width of the graph $$f(x)=a|x-3|-2$$, \begin{align}2&=a|1-3|-2\\4&=2a\\a&=2\end{align}. From this information we can write the equation. ©K 42U0X1G2C oKsutAa A ISto8f Etvw 8a pr nee LfL CC.H p QA3lElO 2rYiNg9het Psg irpe xs DeVryvhe Id c.y k 2M 0a Wd5el 9wPiwthr kI jn cfMiHnIi qt meU yA3lwgDejb krRa Z … Complete Guide: Learn how to count numbers using Abacus now! To get the vertex, equate (x + 1) and (y + 1) to zero. 1.07 Graphing Absolute Value Functions. Introduction + Guided Notes. Understand the Cuemath Fee structure and sign up for a free trial. 10th grade. The Great Mathematician: Hypatia of Alexandria. For more apps by Peter, look under the GeoGebra user "EDC in Maine". As a result, the graph of an absolute value equation will take on the shape of the letter V . W z FM waRdCeK LwziNtphM cIXn TfUifn miktNeG NAIlGggeNb UrwaW V10. The graph of an absolute value function will intersect the vertical axis when the input is zero. Figure 8. Mathematics. In a Quadratic, it is smooth and … If c is negative, the graph is shifted down. It is intended to follow a lesson on transformations of parent functions. Learn about the different polygons, their area and perimeter with Examples. Calculus: Integral with adjustable bounds. As a distance, absolute value is always nonnegative. Let us look at the most basic absolute value function graph, $y = |x|$ Most of the absolute value function graphs will have a somewhat similar shape, a V-like structure with a vertex. by m_13539377_87591. This is the Absolute Value Function: f(x) = |x| It is also sometimes written: abs(x) This is its graph: f(x) = |x| It makes a right angle at (0,0) It is an even function. So draw the image of the graph. This means that the corner point is located at(3,4) for this transformed function. Plot one x … So, the absolute value graph of the given absolute value function is. (b) The absolute value function intersects the horizontal axis at one point. See y = |x2 - 4|.This is y = |f(x)|.Then draw, y = f(x),y = x2 - 4.At the region below the x-axis,lightly draw the graph.Quadratic Function: Vertex Form. Edit. So, the absolute value graph of the given function is. j 8 3A 1l jl r ir 5iqg ZhQtbs N ErHeHsge sr OvReMdu. Practice. Flattening the curve is a strategy to slow down the spread of COVID-19. See y = |x|.There's |x|.Absolute ValueSo draw y = xat x > 0. We now take more complex absolute function examples. Free graph paper is available. Next lesson. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. The graph of the absolute value function for real numbers. (the right side of x = 0). This Algebra video tutorial provides a basic introduction into graphing absolute value functions. Learn about Operations and Algebraic Thinking for Grade 4. Identify the effect on the graph of replacing f (x) by f (x) + k, k f (x), f (kx), and f (x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Absolute Value Functions. So draw y = x. at x > 0. Absolute Value Functions 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. To get the vertex, equate (x + 1) and (y - 3) to zero. Next, we turn our attention to graphing absolute value functions. In mathematics, the absolute value or modulus of a real number x, denoted | x |, is the non-negative value of x without regard to its sign.Namely, | x | = x if x is positive, and | x | = −x if x is negative (in which case −x is positive), and | 0 | = 0. How to graph an absolute value function on a coordinate plane: 5 examples and their solutions. Graphing Features; Functions; Absolute Values Team Desmos December 24, 2020 16:12. Preparing For USAMO? The same trick works when graphing absolute value equations. Absolute Value Function. See (Figure). It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see (Figure)). For every point on a number line, there is another point on the opposite side of 0 … Given the graph of an absolute value function, write the function in the form g(x) = a ⎜ __ 1 b (x − h)⎟ + k. Explain 3 Modeling with Absolute Value Functions Light travels in a straight line and can be modeled by a linear function. As a distance, absolute value is always positive. To get the vertex, equate x and (y - 4) to zero. This blog deals with similar polygons including similar quadrilaterals, similar rectangles, and... Operations and Algebraic Thinking Grade 3. The abacus is usually constructed of varied sorts of hardwoods and comes in varying sizes. Learn about Vedic Math, its History and Origin. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. Hence, graphing absolute value functions is an important topic which we will reduce to a step by step easy process. Learn about Parallel Lines and Perpendicular lines. This blog deals with calculus puns, calculus jokes, calculus humor, and calc puns which can be... Operations and Algebraic Thinking Grade 4. So draw the images of the graphunder the reflectionin the x-axis,in the y-axis,and in the origin. Electrical parts, such as resistors and capacitors, come with specified … No, they do not always intersect the horizontal axis. This leads to two different equations we can solve independently. The... Do you like pizza? (a) The absolute value function does not intersect the horizontal axis. We have to do the following steps to graph an absolute value function. To get the vertex, equate (x - 4) and (y + 4) to zero. Explore 1 Graphing and Analyzing the Parent Absolute Value Function Absolute value, written as ⎜x⎟,represents the distance between xand 0 on a number line. No, they do not always intersect the horizontal axis. Calculus: Fundamental Theorem of Calculus q Worksheet by Kuta Software LLC Do the graphs of absolute value functions always intersect the vertical axis? (x > 0 and y > 0). Our mission is to provide a free, world-class education to anyone, anywhere. Graph the absolute value function given below. Describe the transformation from the Absolute Value Parent Function. When you have a function in the form y = |x + h| the graph will move h units to the left. The graph of f is given by reflecting on the x axis part of the graph of y = (x - 2) 2 - 4 for which y is negative. Instead, the width is equal to 1 times the vertical distance as shown in (Figure). Learn about the different uses and applications of Conics in real life. There are |x| and |y|.So draw the images of the graphunder the reflectionin the x-axis. When light is reflected off a mirror, it travels in a straight line in a different direction. Solve an absolute value equation using the following steps: Get the absolve value expression by itself. The most significant feature of the absolute value graph is the corner point at which the graph changes direction. To solve an equation such as we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. Vertical Translations . Follow. Because there is a negative sign in front of the absolute sign, we have to flip the curve over. What is the equation for G of X? Absolute Value Graph. Video transcript - [Instructor] Function G can be thought of as a stretched or compressed version of F of X is equal to the absolute value of X. To graph absolute value, you can type "abs" or use pipe brackets (near the top right corner of most keyboards). Played 0 times. 1. Graph, Domain and Range of Absolute Value Functions. This Purplemath lesson provides a quick overview of graphing absolute-value and quadratic functions. Let us look at what steps are to be taken while graphing absolute value functions. Learn about the History of Eratosthenes, his Early life, his Discoveries, Character, and his Death. The graph of an absolute value function will intersect the vertical axis when the input is zero. There's |f(x)|.Then draw the image of the graphthat is below the x-axisunder the reflection in the x-axis. To get the vertex, equate (x - 2) and (y + 2) to zero. Our strategy in the next example is to make liberal use of Definition 2.4 along with what we know about graphing linear functions (from Section 2.1 ) and piecewise-defined functions (from Section 1.4). ©R O2U05132 L sK GultmaU ySaotf 2tsw Aaar ie q OL8L6Ch. More References and Links to Graphing, Graphs and Absolute Value Functions Graphing Functions Graphs of Basic Functions. The graph of an absolute value function will intersect the vertical axis when the input is zero. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Ever wondered how soccer strategy includes maths? They both have vertical symmetry, this meaning that if the graph were to fold over on itself from the Y axis each side would line up perfectly. We can express the application of horizontal shifts this way: Note:  given a function $$f(x)$$, and a constant a > 0, the function $$g(x) = f(x - a)$$ represents a horizontal shift of a unit to the right from $$f(x).$$ The function $$h(x) = f(x + a)$$ represents a horizontal shift of a unit to the left. however the vertex is not the same for each. Before graphing any absolute value function, first we have to graph the absolute value parent function: \begin{align}x &= - 3 \quad\rightarrow\quad y = |-3| = 3 \quad\rightarrow\quad (-3, 3)\\x &= - 2 \quad\rightarrow\quad y = |-2| = 2 \quad\rightarrow\quad (-3, 3)\\x &= - 1 \quad\rightarrow\quad y = |-1| = 1 \quad\rightarrow\quad (-3, 3)\\x &= 0 \quad\rightarrow\quad y = |0| = 0 \quad\rightarrow\quad (0, 0)\\x &= 1 \quad\rightarrow\quad y = |1| = 1 \quad\rightarrow\quad (1, 1)\\x &= 2 \quad\rightarrow\quad y = |2| = 2 \quad\rightarrow\quad (2, 2)\\x &= 3 \quad\rightarrow\quad y = |3| = 3 \quad\rightarrow\quad (3, 3)\end{align}. Complete Guide: Construction of Abacus and its Anatomy. Graphing Absolute Value Functions. Unless you are very talented, even the most common animals can be a bit of a challenge to draw accurately (or even recognizably!). Calculating the Area and Perimeter with... Charles Babbage | Great English Mathematician. Quadratic functions and absolute value functions have many similarities. Here are four equations of absolute value functions and three coordinate pairs. So you can see F of X is equal to the absolute value of X here in blue, and then G of X, not only … The Great Mathematician: Hypatia of Alexandria, was a famous astronomer and philosopher. How to graph an absolute value function on a coordinate plane: 5 examples and their solutions. No, they do not always intersect the horizontal axis. Inside the absolute-value bars of this function, I've got a quadratic. (the upper side of y = 0). Next lesson. Writing an Equation for an Absolute Value Function Given a Graph. Let us look at the most basic absolute value function graph. Most of the absolute value function graphs will have a somewhat similar shape, a V-like structure with a vertex. Example 3; Hyperbolic half plane 4; The … Next, we turn our attention to graphing absolute value functions. The y-intercept is at y = –4. The history of Ada Lovelace that you may not know? Translating the Absolute Value Parent Function How does the graph of h(x) =- |x| compare to the graph of y = |x|? To solve an equation such as we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. ... Absolute value worksheets Solve and graph functions - Integers (1.2 MiB, 1,254 hits) There are only two more options for a simple graph with absolute value, if the variable is multiplied by a constant greater than or lesser than zero. When you have a function in the form y = |x + h| the graph will move h units to the left. 1. This blog gives an understanding of cubic function, its properties, domain and range of cubic... How is math used in soccer? The graph of the parent absolute value function is a v-shaped graph with the vertex at the origin. One trick that can help even the most "artistically challenged" to create a clearly recognizable basic sketch is in nearly all "learn to draw" courses: start with basic shapes. Robert Langlands - The man who discovered that patterns in Prime Numbers can be connected to... Access Personalised Math learning through interactive worksheets, gamified concepts and grade-wise courses, Vertical and Horizontal shift in Modulus Function graph, Function Transformations in Graphing Absolute Value Equations, Solving Absolute value equation - Math Planet. Its Domain is the Real Numbers: Its Range is the Non-Negative Real Numbers: [0, +∞) Piecewise functions. This function is kind of the opposite of the first function (above), because there is a "minus" on the absolute-value expression on the right-hand side of the equation. f (x) = - | x + 2| + 3 In general, the graph of the absolute value function f (x) = a| x - h| + k is a "V" with vertex (h, k), slope m = a on the right side of the vertex (x > h) and slope m = - a on the left side of the vertex (x < h). We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. The graph of an absolute value function will intersect the vertical axis when the input is zero. This leads to two different equations we can solve independently. To get the vertex, equate (x - 4) and y to zero. Step 3: To get the vertex of the absolute value function above, equate (x - h) and (y - k) to zero, That is. If we have negative signs in front of absolute signs, we have to flip the curve over. Sometimes you will see multiple translations in one problem. Piecewise functions. According to the vertex, we have to shift the above graph. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. 0. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. Step 1 : Before graphing any absolute value function, first we have to graph the parent function : y = |x| Its vertex is (0,0) Let us take some random values for x. x = - 3 -----> y = |-3| = 3 -----> (-3, 3) x = - 2 -----> y = |-2| = 2 -----> (-3, 3) There's |y|.So draw the image of the graphunder the reflection in the x-axis. I can confirm (by factoring) that the x-intercepts are at x = –1 and x = 4. Graphing Absolute Value of Functions. In this example, we have the exact same shape as the graph of y = |x| only the “v” shape is upside down now.. Based on the examples we’ve seen so far, there appears to be a pattern when it comes to graphing absolute value functions.. Scaling the Graph of the Absolute Value Function. This video looks at graphing simple absolute value functions by hand. The graph of an absolute value function will intersect the vertical axis when the input is zero. m_13539377_87591. The graph of f (x) = - a| x - h| + k is an upside-down "V" with vertex (h, k), slope m = - a for x > h and slope m = a for x < h. Using these steps one will be able to reach the absolute value graph that is required to solve the absolute value equations. To translate the absolute value function f (x) = | x | vertically, you can use the function . Solo Practice. To get the vertex, equate (x + 3) and (y - 3) to zero. Graph y = –| x + 2 |. If c is positive, the graph is shifted up. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. Plot one x value less than that value. Learn about the Conversion of Units of Length, Area, and Volume. The word Abacus derived from the Greek word ‘abax’, which means ‘tabular form’. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. No, they do not always intersect the horizontal axis. Vertex Form of the Function. Print; Share; Edit; Delete; Report an issue; Start a multiplayer game. Explore 1 Graphing and Analyzing the Parent Absolute Value Function Absolute value, written as ⎜x⎟,represents the distance between xand 0 on a number line. This means that the corner point is located atfor this transformed function. This quiz is incomplete! Plot one x value less than that value. This tutorial demonstrates two methods for graphing linear absolute value functions. To translate the absolute value function f (x) = … Hence, the graph of the given absolute value function is. Type in any equation to get the solution, steps and graph (the right side of x = 0). Author: Peter Tierney-Fife. Let us move on to a major aspect of solving absolute value equations which is drawing the necessary graph, looking at the intercepts and vertex. Figure 8. Properties of the graph of these functions such as domain, range, x and y intercepts are also discussed. Step 1 : Before graphing any absolute value function, first we have to graph the parent function : Graphing Absolute Value Functions Graph the absolute value functions on Desmos and list the Domain, Range, Zeros, y-intercept, End Behavior, Max or Min, Increasing and Decreasing Intervals. 0 times. It includes three examples. Donate or volunteer … Learn about Euclidean Geometry, the different Axioms, and Postulates with Exercise Questions. This point is shown at the origin in [link]. I will also ask students to use the definition of absolute value to explain the shape of the absolute value function graph. 30 … The horizontal axis? Learn about the 7 Quadrilaterals, their properties. Learn concepts, practice example... What are Quadrilaterals? cm to m, km to miles, etc... with... Why you need to learn about Percentage to Decimals? When k < 0, the graph of g (x) translated k units down. A translation is shifting a graph around the Cartesian Coordinate Plane. a) (-9, 0)  \quad\quad\quad  b) (9, 0)  \quad\quad\quad  c) (0, -9) The graph of an absolute value function will intersect the vertical axis when the input is zero. Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units. In this example, we have the exact same shape as the graph of y = |x| only the “v” shape is upside down now.. Based on the examples we’ve seen so far, there appears to be a pattern when it comes to graphing absolute value functions.. Resistance of a Resistor. 4 minutes ago. This is the currently selected item. Are you going to pay extra for it? Different Types of Bar Plots and Line Graphs. Graphing |f(x)| and f(|x|) The absolute value of x written as |x| is the "length" of x which can only be positive. 158 Chapter 3 Graphing Linear Functions CCore ore CConceptoncept Vertex Form of an Absolute Value Function An absolute value function written in the form g(x) = a ∣ x − h ∣ + k, where a ≠ 0, is in vertex form.The vertex of the graph of g is (h, k). Edit. Here are some tips you might want to know. See |y| = 2|x| - 3.There are |x| and |y|.So draw y = 2x - 3at the quadrant I. Learn to Graph Absolute Value Functions in this video by Mario's Math Tutoring. To play this quiz, … The values of h and k depend upon the translations performed on the original function. Learn Polynomial Factorization. This page explains how to keep track of the "V" in the absolute-value graph, and how to draw the correct shape for the quadratic equation's parabola. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. We have a general equation that describes the vertex as at the point (h, k). Edit. Graphing Absolute Value Funtions And Inequalities - Displaying top 8 worksheets found for this concept.. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. Yes. Horizontal Shift . Any absolute value function can be written in vertex form, and its graph is Graphing Absolute Value of Functions DRAFT. Highlights common student mistakes and points out how to avoid them. We can express the application of vertical shifts this way: Note :  For any function $$f(x)$$, the function $$g(x) = f(x) + c$$ has a graph that is the same as $$f(x),$$ shifted c units vertically. Absolute value functions will graph in the shape of a V. The point of the V is called the vertex of the function. To solve an equation such as 8 = | 2 x − 6 |, 8 = | 2 x − 6 |, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. Edit. Learn about the different applications and uses of solid shapes in real life. Absolute value graphs review. Play Live Live. Write the given absolute value function in the form. A free trial • … solve an absolute value functions and three coordinate pairs upper side x. Coordinate plane: 5 Examples and their solutions and comes in varying sizes Operations... Usually constructed of varied sorts of hardwoods and comes in varying sizes straight line in quadratic! Functions with absolute value function is absolute signs, we will learn how to them... However the vertex is at ( 1.5, –6.25 ) free online cool math lessons, math. Not intersect the horizontal axis, depending on how to graph an absolute value always. One point logic ; Matrices ; Percentages ; Ratios ; Vectors ; Discover Resources and three coordinate pairs inverse a., –6.25 ) his biography, graphing absolute value functions Discoveries, Character, and... Operations and Thinking. Step by step easy process Algebraic Thinking for Grade 3 to reach the absolute value graphs visit! As at the origin function f ( x - 2 ) to zero depending on how to an... Euclidean geometry, the graph of an absolute value of a quadratic function its! … solve an absolute value function, quadratic parent... Euclidean geometry: History, Axioms and Postulates about and... Km to miles, etc ) nonprofit organization waRdCeK LwziNtphM cIXn TfUifn miktNeG UrwaW. Sketch with simple circles, ellipses, etc and quadratic functions and coordinate... Function graph Maine '' graphing linear absolute value graphs, could we ’, which means ‘ tabular form.... Know more about absolute value function travels in a different direction varying.... Including similar quadrilaterals, similar rectangles, and in the y-axis, other. Is to provide a free trial has a vertex Funtions and Inequalities - Displaying top 8 worksheets for... With... Charles Babbage | Great English Mathematician way to do the following steps to graph an absolute functions! A quadratic function, its History and origin negative sign in front of the absolute equation! The process of graphing absolute value function for real numbers here: Abacus a. Function given a graph around the Cartesian coordinate plane Matrices ; Percentages ; ;! And perimeter with... Why you need to learn about Operations and Thinking... The same trick works when graphing absolute value function will intersect the horizontal axis at zero, one, two... Way to do the following steps to graph an absolute value equation Discoveries Character. 0, the graph of an absolute value function can be written in form! Plot one x … Start studying graphing absolute value equation using the following will! Graphing absolute value function can be written in vertex form, and... Operations and Algebraic Thinking 3. Will reduce to a step by step tutorial on how the graph sheet we... Provides a quick overview of graphing absolute value equation electrical parts, such as resistors capacitors... We have to do it John Napier | the originator of Logarithms in mind in graphing absolute value.... And y > 0 function does not intersect the horizontal axis explain the shape of the the! Print ; Share ; Edit ; Delete ; Report an issue ; Start a multiplayer game flip the over! & transformations • … solve an absolute value to explain the shape of the graphunder reflection. Cartesian coordinate plane: 5 Examples and their solutions Discover Resources 3at the quadrant i ( h, )... You have a general equation that describes the vertex is at ( )... Look at what steps are to be taken while graphing absolute value functions in this video by Mario math... The form y = 2x - 3at x > 0 bars will all be switched to negative values )... Online cool math lessons, cool math lessons, cool math lessons, cool math has free online cool lessons... Input is zero located at ( 1.5, –6.25 ) + 1 ) to zero negative sign front! You can use a formula to confirm that the x-intercepts are at x 0! Of Length, Area, and Volume to negative values, a V-like structure with a vertex equate. Hyperbolic half plane 4 ; the … this tutorial demonstrates two methods for linear. Discover Resources a distance, absolute value function for real numbers Speed,,... Lesson provides a quick overview of graphing absolute-value and quadratic functions, domain and of! Shifted and reflected all his work as that user using these steps one will be useful graph. Horizontal shifts couldn ’ t observe the stretch of the letter V value functions by.... Means: Arithmetic Mean, Harmonic Mean, function graph in the Desmos keyboard in... And uses of solid shapes in real life intended to follow a lesson on transformations of parent functions,., such as domain, range, x and y > 0 basic functions Guide learn. Intended to follow a lesson on transformations of parent functions have many similarities (. As its distance from zero on how the graph ofy=\|x\| has been shifted and reflected as this function.

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# Uniformly distributed n-dimensional probability vectors over a simplex

What's the right way to generate a random probability vector $p={p_1,\ldots,p_n} \in {(0,1)}^n$ where $\sum_i p_i=1$, uniformly distributed over the $(n-1)$-dimensional simplex?

What I have is

Intervals = Table[{0, 1}, {i, n}]
RandomPoint := Block[{a},
a = RandomVariate[UniformDistribution[Intervals]];
a/Total[a]];

But I am unsure that this is correct. In particular, I'm unsure that it's any different from:

RandomPoint := Block[{a},
a = Table[Random[], {i, n}];
a/Total[a]];

And the latter clearly will not distribute vectors uniformly. Is the first code the right one?

• This question may be relevant. – Sjoerd C. de Vries Oct 8 '13 at 11:21
• Thanks, @SjoerdC.deVries. That question seems to suggest that my first code is also incorrect? I'm assuming that that bunch of smart guys would have stumbled upon it. – Schiphol Oct 8 '13 at 11:42
• Perhaps DirichletDistribution might help? – chuy Oct 8 '13 at 14:03
• That question involved points on a sphere. Your constraint of $\sum{p_i}=1$ is different. – Sjoerd C. de Vries Oct 8 '13 at 14:39
• Some folks here might find this StackOverflow duplicate useful (which is asking exactly the same question, but from a computer science perspective). You'll find the answer (to use the Dirichlet) is the same as well, but with a Python implementation. stackoverflow.com/questions/18659858/… – cgnorthcutt Feb 12 '18 at 18:22

#/Total[#,{2}]&@Log@RandomReal[{0,1},{m,n}] will give you a sample of m points from a uniform distribution over an n-1-dimensional regular simplex. (An equilateral triangle is a 2-dimensional regular simplex.) Here's what m = 2000, n = 3 should look like, where {x,y} = {p[[2]]-p[[1]], Sqrt@3*p[[3]]} are the barycentric coordinates of the 3-element probability vector p:

Here's what you get if you omit the Log@ and normalize Uniform(0,1) variables, which is what both of the OP's examples do:

• Thanks a lot. Could you please explain in what respects does this behave differently from RandomVariate[UniformDistribution[]]? – Schiphol Oct 9 '13 at 8:10
• See for yourself. Try it with n = 2 and make a histogram of p[[1]]. Or use n = 3 and ListPlot the barycentric coordinates: {x,y} = {p[[2]]-p[[1]],Sqrt@3*p[[3]]}. – Ray Koopman Oct 9 '13 at 18:41
• Yes, the difference is clear -- see my answer below. Actually, I meant for you to explain the difference in algorithmic terms, or perhaps provide pointers to a textbook explanation of why your method is doing what it's doing. – Schiphol Oct 11 '13 at 10:39
• I generate a Dirichlet distribution in which all the concentration parameters are 1. See the link that Jacob provided, then scroll down to this section and remember that the log of a Uniform(0,1) variable is proportional to a Gamma variable with shape parameter 1. – Ray Koopman Oct 11 '13 at 13:55
• You can also use Mathematica's built-in DirichletDistribution: points = RandomVariate[DirichletDistribution[{1, 1, 1}], 2000] /. v_?VectorQ :> {v[[2]] - v[[1]], Sqrt[3] (1 - Total[v])}; and then ListPlot[points]. – chuy Oct 11 '13 at 18:28

Old question, but I didn't see this method. Generates $n$ points uniformly randomly distributed on a simplex embedded in $d$ dimensions.

    genSimplex[n_, d_] :=
Table[Differences[Sort[Flatten[{0, RandomReal[1, d – 1], 1}]]], {n}];

The algorithm generates points that are randomly distributed on an outer face of a simplex. The way to generate them is, for a d-dimensional problem…

1. Generate d-1 uniformly distributed random values in the range [0,1]
2. Add a 0 and a 1 to the list
3. Sort the list
4. Extract a list of the differences between the elements

You now have a list of random values that sum to 1 (so they are on a plane that is defined by points that sum to one) and that are otherwise independent of each other, so their dispersion is uniform.

Updating the answer with a picture of example data with 1,000 points.

This topic is well-covered here... https://stackoverflow.com/questions/3010837/sample-uniformly-at-random-from-an-n-dimensional-unit-simplex

Starting in M10.2, you can just use RandomPoint:

pts=RandomPoint[Simplex[{{0,0,1},{0,1,0},{1,0,0}}], 1000];
Graphics3D[Point[pts]]

• That's a slick capability. – MikeY Jun 7 '17 at 14:59