Solved Examples: Vertical and Horizontal Stretches and Compressions of Functions

📚 Quick Study Guide

• 📏 Vertical Stretch: Multiplying a function $f(x)$ by a constant $a > 1$ stretches the graph vertically. The new function is $af(x)$.
• 📉 Vertical Compression: Multiplying a function $f(x)$ by a constant $0 < a < 1$ compresses the graph vertically. The new function is $af(x)$.
• ↔️ Horizontal Compression: Replacing $x$ with $bx$ where $b > 1$ compresses the graph horizontally. The new function is $f(bx)$.
• 📈 Horizontal Stretch: Replacing $x$ with $bx$ where $0 < b < 1$ stretches the graph horizontally. The new function is $f(bx)$.
• ➕ Key Point: Vertical transformations affect the y-values, while horizontal transformations affect the x-values. Remember that horizontal transformations act in the opposite way you might initially think!

Practice Quiz

1. What transformation does the function $3f(x)$ represent?
1. Vertical compression by a factor of 3
2. Horizontal stretch by a factor of 3
3. Vertical stretch by a factor of 3
4. Horizontal compression by a factor of 3


2. What transformation does the function $f(2x)$ represent?
1. Horizontal stretch by a factor of 2
2. Vertical stretch by a factor of 2
3. Horizontal compression by a factor of 2
4. Vertical compression by a factor of 2


3. What transformation does the function $\frac{1}{2}f(x)$ represent?
1. Vertical stretch by a factor of 2
2. Horizontal compression by a factor of $\frac{1}{2}$
3. Vertical compression by a factor of $\frac{1}{2}$
4. Horizontal stretch by a factor of 2


4. What transformation does the function $f(\frac{1}{3}x)$ represent?
1. Horizontal compression by a factor of $\frac{1}{3}$
2. Vertical compression by a factor of $\frac{1}{3}$
3. Horizontal stretch by a factor of 3
4. Vertical stretch by a factor of 3


5. If the point (2, 4) lies on the graph of $f(x)$, what point must lie on the graph of $2f(x)$?
1. (1, 4)
2. (2, 2)
3. (2, 8)
4. (4, 4)


6. If the point (6, 3) lies on the graph of $f(x)$, what point must lie on the graph of $f(3x)$?
1. (2, 3)
2. (18, 3)
3. (6, 9)
4. (6, 1)


7. The graph of $y = f(x)$ is vertically compressed by a factor of $\frac{1}{4}$. Which of the following represents the transformed function?
1. $y = 4f(x)$
2. $y = f(4x)$
3. $y = \frac{1}{4}f(x)$
4. $y = f(\frac{1}{4}x)$