Apply transformations (shift, stretch/compress, reflection) to nonlinear graphs and equations. | Nonlinear functions | SAT Math

Moving, stretching, compressing, and flipping graphs using algebraic changes to the equation.

Core Idea

For $𝑦 = 𝑎 \cdot 𝑓 (𝑥 - ℎ) + 𝑘$ : $ℎ$ shifts right, $𝑘$ shifts up, $| 𝑎 | > 1$ stretches vertically, $0 < | 𝑎 | < 1$ compresses vertically, and a negative $𝑎$ reflects over the x-axis. Changes inside the function (to $𝑥$ ) move the graph opposite to the sign; changes outside move it in the same direction as the sign.

Understanding

Transformation questions test whether you know the rules and — critically — the direction each change moves the graph.

Vertical shifts: $𝑓 (𝑥) + 𝑘$ moves the graph up by $𝑘$ (down if $𝑘 < 0$ ).

Horizontal shifts: $𝑓 (𝑥 - ℎ)$ moves the graph right by $ℎ$ . The direction is opposite the sign inside the parentheses: $𝑓 (𝑥 - 3)$ shifts right 3, $𝑓 (𝑥 + 2)$ shifts left 2.

Vertical stretch/compress: $𝑎 \cdot 𝑓 (𝑥)$ stretches the graph vertically if $| 𝑎 | > 1$ and compresses it if $0 < | 𝑎 | < 1$ .

Reflection: $- 𝑓 (𝑥)$ reflects over the x-axis (flips upside down). $𝑓 (- 𝑥)$ reflects over the y-axis (flips left-right).

When multiple transformations appear in one equation, apply them in order: horizontal shift, stretch/compress, reflection, then vertical shift.

Step by Step

Identify the parent function (e.g., $𝑥 2$ , $2 𝑥$ , $| 𝑥 |$ ).
Read the horizontal shift from inside the function argument: $𝑓 (𝑥 - ℎ)$ → shift right $ℎ$ .
Read the vertical stretch/compression and reflection from the coefficient: $𝑎 \cdot 𝑓 (\dots)$ .
Read the vertical shift from the constant added outside: $\dots + 𝑘$ .
Apply transformations to key points of the parent graph to sketch or analyze the new graph.

Misconceptions

Shifting in the wrong horizontal direction: $𝑓 (𝑥 + 3)$ shifts left, not right.
Confusing vertical and horizontal stretches — a coefficient outside affects y-values (vertical); one inside (like $𝑓 (2 𝑥)$ ) affects x-values (horizontal, and by the reciprocal).
Applying transformations in the wrong order, especially mixing up vertical shift and stretch.

Question

Worked Example

The function $𝑔$ is defined by $𝑔 (𝑥) = (𝑥 + 3) 2 - 5$ . The graph of $𝑔$ is the graph of $𝑦 = 𝑥 2$ shifted in which of the following ways?

Select an answer to see the explanation