Question

How do you find absolute value equation from graph?

Answer 1

To find an equation in the form `y = aabs(x-h) +k`, do the following:

Explanation:

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

The vertex is the point `(h, k)`, 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 `(h,k)` find another point on the graph to the right of the vertex. Call it `(x_2, y_2)`.

Find the slope of the line through the points `(h,k)` and `(x_2, y_2)`. That is `a`

`a = (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 `(3,2)` so the equation looks like

`y = aabs(x-3)+2`

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

`a` is the slope:

`a = (7-2)/(5-3)`

So `a = 5/2`

The equation is:

`y = 5/2abs(x-3)+2`

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

`2y = 5abs(x-3)+2`

Example 2

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

Find the vertex:

.
.

.

The vertex is `(4,1)`.

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

.

.

.I'll use the point `(6, -3)`

`a = (-3-1)/(6-4) = (-4)/2 = -2`

The equation is

`y = -2abs(x-4)+1`