Transforming Functions

This is a Functions and Graphs topic. The SAT might ask:

This is called transforming a function — you’re changing how the graph looks without changing the basic shape.

The 4 Main Transformations

The four main transformations of a function are vertical shift, horizontal shift, reflection, and dilation

If you add or subtract outside the function:

$f(x)+k$

Example:
If $f(x)=x^2$ , then $f(x)+3=x^2+3$ is 3 units higher.

If you add or subtract inside the function:

$f(x+h)$

Example:

If $f(x)=x^2$ , then $f(x-2)=(x-2)^2$ is horizontal shift 2 units right.

Tip: Horizontal shifts feel backward.

Negating the function or the $x$ reflects a function.

Example:

If $f(x)=x^3$ , then $-f(x)=-x^3$ is a flip over the y-axis.

A dilation is when the entire function is multiplied by a constant (other than 1 or 0).

$a f(x)$

Example:

If $f(x)=x^2$ , then

$3f(x)=3x^2$ results in a vertical stretch and it will look skinnier
$\dfrac{1}{2}f(x)=\dfrac{1}{2}x^2$ results in a vertical compression and it will look wider

Given $f(x)$ and $g(x)=f(x+1)-5$ , how does the graph of $g(x)$ compare to the graph of $f(x)$ ?

The graph of $g(x)$ is the same as the graph of $f(x)$ but it is translated left 1 unit and down 5 units.

You don’t have to graph everything — just understand the effect.
If they give you the graph of $f(x)$ and ask for $f(x-2)+1$ , apply the shifts step-by-step.
Use Desmos! You can type in functions and quickly see how the transformations affect them.