Before you read

Answer first — see what you already know.

If a temperature curve is $g (x) = f (x) + 5$ , how does $g$ relate to $f$ ?

A
Shifted up 5 units
B
Shifted right 5 units
C
Stretched vertically by 5
D
Shifted down 5 units

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A growth curve $g (x) = f (x - 4)$ is a transformation of $f (x)$ . Which direction is the shift?

A
Shifted left 4 units
B
Shifted right 4 units
C
Shifted down 4 units
D
Reflected over $x = 4$

Check your answer

Answer:

Theory

Transformation Rules

Starting from $y = f(x)$ , here's how each transformation changes the graph:

Parent function y = x²

Transformation	Equation	Effect
Shift up $k$	$f(x) + k$	Every point moves up $k$ units
Shift down $k$	$f(x) - k$	Every point moves down $k$ units
Shift right $h$	$f(x - h)$	Every point moves right $h$ units
Shift left $h$	$f(x + h)$	Every point moves left $h$ units
Reflect over x-axis	$-f(x)$	Flip upside down
Reflect over y-axis	$f(-x)$	Flip left-right
Vertical stretch	$a \cdot f(x)$ , $\|a\| > 1$	Taller/narrower
Vertical compression	$a \cdot f(x)$ , $0 < \|a\| < 1$	Shorter/wider

y = (x-3)² + 2: shifted right 3, up 2

Key insight: changes inside $f(\text{here})$ affect $x$ (horizontal, opposite direction). Changes outside affect $y$ (vertical, same direction).

Theory

Combining Transformations

Most SAT questions combine 2-3 transformations:

$g(x) = a \cdot f(x - h) + k$

- $a$ : vertical stretch/compression and possible reflection
- $h$ : horizontal shift (right if positive)
- $k$ : vertical shift (up if positive)

Order matters: apply horizontal shift first, then stretch, then vertical shift.

Example 1

If $f(x) = x^2$ , describe the transformation to get $g(x) = 2(x - 3)^2 + 1$ .

Start with $y = x^2$ .

Shift right 3: $y = (x-3)^2$

Vertical stretch by 2: $y = 2(x-3)^2$

Shift up 1: $y = 2(x-3)^2 + 1$

Right 3, vertical stretch by factor 2, up 1.

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2 more sections including examples, practice problems, and step-by-step solutions.

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