SAT — mathHard4 min read

Function Transformations

Master translations, reflections, stretches, and compressions of function graphs for the Digital SAT.

Tutor AI Team2026-02-25T01:25:39.011Z

Function transformations modify the graph of a function by shifting, reflecting, stretching, or compressing it. The Digital SAT tests your ability to predict how changes to the equation affect the graph. Understanding transformations helps with quadratic, absolute value, exponential, and other function types.

Core Concepts

Vertical Translations

$f(x) + k$ : shifts graph up by $k$ units (if $k > 0$ ) or down (if $k < 0$ ).

Horizontal Translations

$f(x - h)$ : shifts graph right by $h$ units. $f(x + h)$ : shifts graph left by $h$ units.

Note: The horizontal direction is opposite to the sign inside the function.

Vertical Stretch/Compression

$a \cdot f(x)$ :

$|a| > 1$ : vertical stretch (taller)
$0 < |a| < 1$ : vertical compression (shorter)

Horizontal Stretch/Compression

$f(bx)$ :

$|b| > 1$ : horizontal compression (narrower)
$0 < |b| < 1$ : horizontal stretch (wider)

Reflections

$-f(x)$ : reflection over the x-axis (flip vertically). $f(-x)$ : reflection over the y-axis (flip horizontally).

Combined Transformations

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

Order of application (inside-out):

Horizontal shift by $h$
Horizontal stretch/compression by $\frac{1}{b}$
Vertical stretch/compression by $a$
Vertical shift by $k$

Transformation Summary Table

Change to equation	Effect on graph
$f(x) + k$	Up $k$
$f(x) - k$	Down $k$
$f(x - h)$	Right $h$
$f(x + h)$	Left $h$
$-f(x)$	Reflect over x-axis
$f(-x)$	Reflect over y-axis
$af(x)$ , $a>1$	Vertical stretch
$af(x)$ , $0<a<1$	Vertical compression

Strategy Tips

Tip 1: Inside = Horizontal (Opposite Direction)

Changes inside the function argument ( $x$ part) affect the graph horizontally, and they work in the opposite direction of what you'd expect.

Tip 2: Outside = Vertical (Same Direction)

Changes outside the function affect the graph vertically, in the same direction as the sign.

Tip 3: Track the Vertex/Key Point

For parabolas, apply the transformation to the vertex. For other functions, track a key point.

Tip 4: Match Before and After

If the SAT shows the original and transformed graph, identify which transformation(s) occurred by comparing key features.

Tip 5: Use Desmos to Experiment

Type $f(x) = x^2$ and then $f(x-3) + 2$ to see the effect in real time.

Worked Example: Example 1

Problem

If $f(x) = x^2$ , what is the graph of $g(x) = (x-3)^2 + 2$ ?

The graph of $f(x) = x^2$ shifted right 3 and up 2. Vertex moves from $(0,0)$ to $(3,2)$ .

Solution

Worked Example: Example 2

Problem

If $f(x) = |x|$ , describe $g(x) = -|x + 1| + 4$ .

$|x+1|$ : shift left 1
$-|x+1|$ : reflect over x-axis (opens downward)
$-|x+1| + 4$ : shift up 4

Vertex moves from $(0,0)$ to $(-1, 4)$ . Opens downward.

Solution

Worked Example: SAT-Style

Problem

The graph of $y = f(x)$ is shown. Which equation represents the graph after it is shifted 2 units left and 5 units down?

$y = f(x + 2) - 5$

Solution

Worked Example: Example 4

Problem

If $g(x) = 3f(x)$ , how does $g$ compare to $f$ ?

Vertical stretch by factor 3. Every y-value is tripled. The graph is taller/steeper.

Solution

Worked Example: Example 5

Problem

The graph of $y = f(x)$ passes through $(2, 5)$ . What point must be on $y = f(x - 4) + 1$ ?

Shift right 4 and up 1: $(2 + 4, 5 + 1) = (6, 6)$ .

Solution

Practice Problems

Problem 1

If $f(x) = x^2$ , write the equation of $f$ shifted right 5 and down 3.

Problem 2

Describe the transformation from $f(x)$ to $-f(x-2)$ .

Problem 3

The vertex of $f(x) = x^2$ is at $(0, 0)$ . Where is the vertex of $g(x) = 2(x+1)^2 - 7$ ?

Problem 4

If $(3, 8)$ is on $y = f(x)$ , what point is on $y = f(-x)$ ?

Problem 5

Which is narrower: $y = x^2$ or $y = 4x^2$ ?

Problem 6

If $f(x) = \sqrt{x}$ , write the equation that reflects $f$ over the x-axis and shifts it up 6.

Want to check your answers and get step-by-step solutions?

Common Mistakes

Getting horizontal direction backwards. $f(x-3)$ shifts RIGHT, not left.
Confusing vertical and horizontal transformations. Inside changes → horizontal. Outside → vertical.
Applying transformations in wrong order. Apply inside (horizontal) first, then outside (vertical).
Forgetting that $-f(x)$ and $f(-x)$ are different. One flips vertically, the other horizontally.
Not tracking specific points. If a specific point is given, apply the transformation to its coordinates.

Key Takeaways

✓
Vertical shifts: $f(x) \pm k$ (up/down).
✓
Horizontal shifts: $f(x \mp h)$ (right/left — opposite of sign).
✓
Reflections: $-f(x)$ = flip over x-axis; $f(-x)$ = flip over y-axis.
✓
Stretches: $af(x)$ with $|a| > 1$ stretches vertically.
✓
Inside changes are horizontal and opposite; outside changes are vertical and direct.
✓
Track a key point or vertex through the transformation.

Tags:

#sat#math#advanced-math#functions#transformations

Ready to Ace Your SAT math?

Get instant step-by-step solutions to any problem. Snap a photo and learn with Tutor AI — your personal exam prep companion.

Core Concepts

Vertical Translations

Horizontal Translations

Vertical Stretch/Compression

Horizontal Stretch/Compression

Reflections

Combined Transformations

Transformation Summary Table

Strategy Tips

Tip 1: Inside = Horizontal (Opposite Direction)

Tip 2: Outside = Vertical (Same Direction)

Tip 3: Track the Vertex/Key Point

Tip 4: Match Before and After

Tip 5: Use Desmos to Experiment

Worked Example: Example 1

Worked Example: Example 2

Worked Example: SAT-Style

Worked Example: Example 4

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?