Standard Form Absolute Value Equation

[Jason76]

Is there such an equation? I know such equations exist for parabolas and circles.

Hypothetical Absolute Value Equation:

$\displaystyle y = \mid x - h \mid + k$

For vertex:

h = x

k = y

So

$\displaystyle y = \mid x - 3 \mid + 1$

or

$\displaystyle y = \mid x - (+3) \mid + 1$ (under the hood)

Would be the graph of the an absolute function with it's vertex at

$\displaystyle (3,1)$

Keeping in mind that the absolute value equation (with vertex at $\displaystyle (0,0)$) is:

$\displaystyle y = \mid x \mid$

with an upside down triangle shaped graph (which can be shifted by changing the vertex).

 

[soroban]

Hello, Jason76!

You are correct! .Nice thinking process!


$\displaystyle y \,=\,|x|$ is a $\displaystyle \vee$-shaped graph with its vertex at $\displaystyle (0.0).$


The general form might be: .$\displaystyle y \:=\:a|x-h| + k$
. . where the vertex is $\displaystyle (h,k)$
. . and $\displaystyle a$ modifies the "spread" of the $\displaystyle \vee.$

 

[Jason76]

h:shifts function left or right

k: shifts function up or down

a: determines width on parabola and absolute value functions.

h or k only affects the vertex coordinates if the function has a vertex. Otherwise, each value (that you know) on the function has to be moved left, right, up, or down based on h and k.

Some different types of standard functon equations:

Absolute Value:

$\displaystyle y = \mid x \mid$

$\displaystyle y = a\mid x \mid$

$\displaystyle y = a\mid x - h \mid + k$

Square Root:

$\displaystyle y = \sqrt{x}$

$\displaystyle y = a\sqrt{x}$

$\displaystyle y - a\sqrt{x -h} + k$

Common Slanted Line:

$\displaystyle y = (x)$

$\displaystyle y = a(x)$

$\displaystyle y = a(x - h) + k$

Vertical Parabola

$\displaystyle y = (x)^{2}$

$\displaystyle y = a(x)^{2}$

$\displaystyle y = a(x - h)^{2} + k$