# How do you find absolute value equation from graph?

Jul 28, 2015

To find an equation in the form $y = a \left\mid x - h \right\mid + k$, do the following:

#### Explanation:

The graph will be shaped like a $V$ or an upside down $V$

The vertex is the point $\left(h , k\right)$, so look at the graph to determine the coordinates of the vertex. (It is the point of the $V$.)

The graph will have straight lines on both sides of the vertex. To the right of $x = h$, the slope of the line will be what we need for $a$ in the equation. (To the left of $x = h$, the slope will be $- a$.

So after you have found the vertex $\left(h , k\right)$ find another point on the graph to the right of the vertex. Call it $\left({x}_{2} , {y}_{2}\right)$.

Find the slope of the line through the points $\left(h , k\right)$ and $\left({x}_{2} , {y}_{2}\right)$. That is $a$

$a = \frac{{y}_{2} - k}{{x}_{2} - h}$

Here are two examples:

Example 1

graph{y = 5/2abs(x-3) + 2 [-3.9, 16.1, -0.856, 9.145]}

(Use your mouse: wheel to scroll in or out and click, hold and drag the graph around as needed.)

The vertex is at $\left(3 , 2\right)$ so the equation looks like

$y = a \left\mid x - 3 \right\mid + 2$

To find $a$, find a pont on the graph to the right of the vertex. I'll use $\left(5 , 7\right)$:

$a$ is the slope:

$a = \frac{7 - 2}{5 - 3}$

So $a = \frac{5}{2}$

The equation is:

$y = \frac{5}{2} \left\mid x - 3 \right\mid + 2$

If you want to get rid of the fraction, multiply both sides by $2$, to get:

$2 y = 5 \left\mid x - 3 \right\mid + 2$

Example 2

graph{y = -2abs(x-4)+1 [-1.25, 11.24, -3.97, 2.276]}

Find the vertex:

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.

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The vertex is $\left(4 , 1\right)$.

Find $a$.
First find a point to the right of the vertex, then $a$ = the slope of the line throught the two points.

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.I'll use the point $\left(6 , - 3\right)$

$a = \frac{- 3 - 1}{6 - 4} = \frac{- 4}{2} = - 2$

The equation is

$y = - 2 \left\mid x - 4 \right\mid + 1$