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

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

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

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