Graphing Examples for Function Stretches and Compressions in Algebra 2

📚 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 Stretch: Replacing $x$ with $\frac{x}{b}$ where $b > 1$ stretches the graph horizontally. The new function is $f(\frac{x}{b})$.
• Narrower ➡️ Horizontal Compression: Replacing $x$ with $bx$ where $b > 1$ compresses the graph horizontally. The new function is $f(bx)$.
• ➕ Key Point: Stretches and compressions change the shape of the graph, but they don't shift it.

Practice Quiz

1. Which transformation represents a vertical stretch of the function $f(x)$ by a factor of 3?

   1. $f(3x)$
   2. $\frac{1}{3}f(x)$
   3. $3f(x)$
   4. $f(\frac{x}{3})$

2. What transformation does the equation $g(x) = f(2x)$ represent?

   1. Vertical stretch by a factor of 2
   2. Horizontal stretch by a factor of 2
   3. Vertical compression by a factor of 2
   4. Horizontal compression by a factor of 2

3. If $f(x) = x^2$, which of the following represents a vertical compression by a factor of $\frac{1}{2}$?

   1. $g(x) = 2x^2$
   2. $g(x) = \frac{1}{2}x^2$
   3. $g(x) = (2x)^2$
   4. $g(x) = (\frac{1}{2}x)^2$

4. The graph of $y = f(x)$ is transformed to $y = f(\frac{1}{3}x)$. This transformation represents:

   1. Horizontal compression by a factor of 3
   2. Vertical compression by a factor of 3
   3. Horizontal stretch by a factor of 3
   4. Vertical stretch by a factor of 3

5. Which of the following transformations will make the graph of $f(x) = |x|$ wider?

   1. $f(2x)$
   2. $2f(x)$
   3. $\frac{1}{2}f(x)$
   4. $f(\frac{1}{2}x)$

6. What is the effect on the graph of $f(x)$ when transformed to $3f(\frac{1}{2}x)$?

   1. Vertical compression by 3, horizontal compression by 2
   2. Vertical stretch by 3, horizontal stretch by 2
   3. Vertical stretch by 3, horizontal compression by 2
   4. Vertical compression by 3, horizontal stretch by 2

7. If the point (2, 4) lies on the graph of $y = f(x)$, what point must lie on the graph of $y = 2f(x)$?

   1. (1, 4)
   2. (2, 2)
   3. (4, 4)
   4. (2, 8)