Function Transformations

Q: How can I study Function Transformations effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Q: Is this Function Transformations study guide free?

Yes — all study notes, flashcards, and practice problems for Function Transformations on Study Mondo are free to access. No account is needed.

Q: What course covers Function Transformations?

Function Transformations is part of the AP Precalculus course on Study Mondo, specifically in the Function Fundamentals section. You can explore the full course for more related topics and practice resources.

Q: Are there practice problems for Function Transformations?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Understanding how to shift, stretch, compress, and reflect functions

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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Function Transformations

Types of Transformations

Given a parent function $f(x)$ , we can transform it in several ways:

Vertical Transformations

Vertical Shift: $f(x) + k$ shifts the graph up by $k$ units (down if $k < 0$ )
Vertical Stretch/Compression: $a \cdot f(x)$ $a \cdot f (x)$
- If $|a| > 1$ : vertical stretch
- If $0 < |a| < 1$ : vertical compression
- If $a < 0$ : reflection across x-axis

Horizontal Transformations

Horizontal Shift: $f(x - h)$ shifts the graph right by $h$ units (left if $h < 0$ )
Horizontal Stretch/Compression: $f(bx)$ $f (b x)$
- If $|b| > 1$ : horizontal compression (faster)
- If $0 < |b| < 1$ : horizontal stretch (slower)
- If $b < 0$ : reflection across y-axis

General Form

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

Where:

$a$ : vertical stretch/compression and reflection
$b$ : horizontal stretch/compression and reflection
$h$ : horizontal shift
$k$ : vertical shift

Order of Transformations

Horizontal shifts and stretches (inside the function)
Vertical stretches and reflections (coefficient)
Vertical shifts (outside)

Key Points to Remember

Inside changes ( $x$ transformations) work opposite to intuition
$f(x - 3)$ shifts RIGHT, not left
$f(2x)$ compresses horizontally, not stretches

📚 Practice Problems

1Problem 1easy

❓ Question:

The graph of $f(x) = x^2$ is transformed to $g(x) = 2(x-3)^2 + 1$ . Describe all transformations applied.

💡 Show Solution

Identify each transformation:

Starting with $f(x) = x^2$ , we have $g(x) = 2(x-3)^2 + 1$

Compare to $g(x) = a \cdot f(b(x - h)) + k$ :

$a = 2$ : vertical stretch by factor of 2
$b = 1$ : no horizontal stretch
$h = 3$ : horizontal shift right 3 units
$k = 1$ : vertical shift up 1 unit

Transformations in order:

Shift right 3 units
Stretch vertically by factor of 2
Shift up 1 unit

Vertex: The vertex of $f(x) = x^2$ is at $(0, 0)$ . After transformations, the new vertex is at $(3, 1)$ .

2Problem 2medium

❓ Question:

Given $f(x) = \sqrt{x}$ , write the equation for the function that results from: reflecting across the x-axis, shifting left 2 units, and shifting down 3 units.

💡 Show Solution

Apply transformations step by step:

Starting with $f(x) = \sqrt{x}$

Step 1: Reflect across x-axis

Multiply by -1: $-f(x) = -\sqrt{x}$

Step 2: Shift left 2 units

Replace $x$ with $x + 2$ : $-f(x+2) = -\sqrt{x+2}$

Step 3: Shift down 3 units

Subtract 3: $-f(x+2) - 3 = -\sqrt{x+2} - 3$

Final answer: $g(x) = -\sqrt{x+2} - 3$

Domain: Since we need $x + 2 \geq 0$ , the domain is $x \geq -2$ or $[-2, \infty)$

Range: Since $\sqrt{x+2} \geq 0$ , we have $-\sqrt{x+2} \leq 0$ , so $-\sqrt{x+2} - 3 \leq -3$ . Range is $(-\infty, -3]$

3Problem 3hard

❓ Question:

The point $(4, -2)$ lies on the graph of $y = f(x)$ . What point must lie on the graph of $y = -3f(\frac{1}{2}x + 1) + 5$ ?

💡 Show Solution

Work backwards from the transformed point:

We know $(4, -2)$ is on $y = f(x)$ , so $f(4) = -2$ .

For $g(x) = -3f(\frac{1}{2}x + 1) + 5$ , we need to find which $x$ makes the inside equal to 4.

Find the x-coordinate: Set $\frac{1}{2}x + 1 = 4$

Solve for $x$ : $\frac{1}{2}x = 3$ $x = 6$

Find the y-coordinate: When $x = 6$ : $g(6) = -3f(\frac{1}{2}(6) + 1) + 5$ $= -3f(4) + 5$ $= -3(-2) + 5$ $= 6 + 5 = 11$

Answer: The point $(6, 11)$ must lie on $y = -3f(\frac{1}{2}x + 1) + 5$

Verification of transformations:

Original point: $(4, -2)$
Horizontal: $\frac{1}{2}x + 1 = 4$ means shift left 2 then stretch by 2: $4 \to 2 \to 6$
Vertical: $-2 \to 6$ (multiply by -3) $\to 11$ (add 5) ✓

4Problem 4medium

❓ Question:

Describe the transformations from f(x) = x² to g(x) = -2(x + 3)² - 5.

💡 Show Solution

Step 1: Identify each transformation in order: g(x) = -2(x + 3)² - 5

Step 2: Horizontal shift: (x + 3) means shift LEFT 3 units

Step 3: Vertical stretch/reflection: Coefficient of -2: • Factor of 2: vertical stretch by factor of 2 • Negative sign: reflection over x-axis

Step 4: Vertical shift: -5 means shift DOWN 5 units

Step 5: Order of transformations:

Start with f(x) = x²
Shift left 3 units: (x + 3)²
Stretch vertically by 2: 2(x + 3)²
Reflect over x-axis: -2(x + 3)²
Shift down 5 units: -2(x + 3)² - 5

Answer: Shift left 3, stretch vertically by 2, reflect over x-axis, shift down 5

5Problem 5hard

❓ Question:

If f(x) = √x, write the equation for the function that results from compressing f horizontally by a factor of 4, then shifting right 2 units and up 1 unit.

💡 Show Solution

Step 1: Start with f(x) = √x

Step 2: Horizontal compression by factor of 4: Replace x with 4x: f(4x) = √(4x)

Step 3: Shift right 2 units: Replace x with (x - 2): √(4(x - 2))

Step 4: Shift up 1 unit: Add 1: √(4(x - 2)) + 1

Step 5: Simplify if desired: g(x) = √(4(x - 2)) + 1 = 2√(x - 2) + 1

Step 6: Verify: • √(4x) compresses horizontally by 4 • √(4(x-2)) shifts right 2 • Adding 1 shifts up 1 ✓

Answer: g(x) = 2√(x - 2) + 1 or √(4(x - 2)) + 1

Explain using:

⚠️ Common Mistakes: Function Transformations

Avoid these 4 frequent errors

🌍 Real-World Applications: Function Transformations

See how this math is used in the real world

📝 Worked Example: Related Rates — Expanding Circle

Problem:

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

2Write the relationship between variables

3Differentiate both sides with respect to time

4Substitute known values

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Composite and Inverse Functions

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❓ Frequently Asked Questions

What is Function Transformations?▾

Understanding how to shift, stretch, compress, and reflect functions

How can I study Function Transformations effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Function Transformations study guide free?▾

Yes — all study notes, flashcards, and practice problems for Function Transformations on Study Mondo are free to access. No account is needed.

What course covers Function Transformations?▾

Function Transformations is part of the AP Precalculus course on Study Mondo, specifically in the Function Fundamentals section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Function Transformations?▾

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Function Transformations

Try the Interactive Version!

Function Transformations

Types of Transformations

Vertical Transformations

Horizontal Transformations

General Form

Order of Transformations

Key Points to Remember

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2medium

❓ Question:

3Problem 3hard

❓ Question:

4Problem 4medium

❓ Question:

5Problem 5hard

❓ Question:

⚠️ Common Mistakes: Function Transformations

🌍 Real-World Applications: Function Transformations

📝 Worked Example: Related Rates — Expanding Circle

Practice Lab

Practice with Flashcards

Browse All Topics

📌 Related Topics in Function Fundamentals

❓ Frequently Asked Questions