Transformations of Functions

You’ll be able to

Translate a graph horizontally and vertically from the equation
Identify reflections across the x-axis and y-axis
Combine shifts and reflections in the correct order

Shifts: inside moves x, outside moves y

Starting from a parent function f(x), you can slide its graph without changing its shape. Changes outside the function move it vertically: f(x) + k shifts up by k (down if k is negative). Changes inside, attached to x, move it horizontally — and they run backward: f(x − h) shifts right by h, while f(x + h) shifts left. So g(x) = f(x − 4) + 2 is f shifted right 4 and up 2.

Reflections: a minus sign flips the graph

A negative sign creates a mirror image. −f(x) negates every output, flipping the graph across the x-axis (up becomes down). f(−x) negates every input, flipping the graph across the y-axis (left becomes right). Watching where the minus sign sits — outside for a vertical flip, inside for a horizontal flip — tells you which axis is the mirror.

Transformation summary

g(x) = a·f(x − h) + k

h shifts horizontally (right for −h inside), k shifts vertically, a scales/reflects vertically (a < 0 flips across the x-axis). A minus on x inside, f(−x), reflects across the y-axis.

Worked example

Describe how g(x) = f(x − 4) + 2 transforms the graph of f(x).

1.Look inside the function: the input is x − 4. An inside subtraction shifts horizontally the opposite way, so the graph moves right by 4.
2.Look outside the function: + 2 is added to the whole output, shifting the graph up by 2.
3.There is no negative sign and no coefficient other than 1, so there is no reflection or stretch.
4.Combine: the graph of f moves 4 units right and 2 units up.

Answer: g(x) = f(x − 4) + 2 shifts the graph of f right 4 units (inside subtraction) and up 2 units (outside addition), with no reflection or stretch.

Checkpoint

The graph of g(x) = f(x − 4) + 2 is obtained from f(x) by which transformation?

Watch out

Horizontal shifts run counterintuitively: f(x − 4) moves right, not left, because you need a larger x to reach the same input value. Trust the rule, not the sign at face value.

Checkpoint

Which transformation does g(x) = −f(x) represent?

On the exam

When several transformations combine, handle inside changes (horizontal shifts, y-axis flips) separately from outside changes (vertical shifts, x-axis flips). Describe each precisely — the AP rubric awards the direction and the axis, not just the word “shift.”

Answer the 2 checkpoints as you read.