Question

in exercises 9 - 12, use the graphs of f and h to describe the transformation from the graph of f to the graph of h. (see example 2.) 9. f(x)=\frac{2}{3}x + 4, h(x)=-f(x) 10. f(x)=-3x + 1, h(x)=f(-x) 11. f(x)=-5 - x, h(x)=f(-x) 12. f(x)=\frac{1}{4}x - 2, h(x)=-f(x) in exercises 13 - 18, use the graphs of f and r to describe the transformation from the graph of f to the graph of r. (see example 3.) 13. f(x)=\frac{3}{2}x - 1, r(x)=2f(x) 14. f(x)=-x, r(x)=f(\frac{1}{4}x) 15. f(x)=-2x - 4, r(x)=f(\frac{1}{2}x) 16. f(x)=3x + 5, r(x)=f(\frac{1}{3}x) 17. f(x)=\frac{2}{3}x + 1, r(x)=3f(x) 18. f(x)=-\frac{1}{4}x - 2, r(x)=4f(x) in exercises 19 - 24, use the graphs of f and h to describe the transformation from the graph of f to the graph of h. (see example 4.) 19. f(x)=x + 2, h(x)=f(3x) 20. f(x)=-2x - 6, h(x)=\frac{1}{3}f(x) 21. f(x)=3x - 12, h(x)=\frac{1}{6}f(x) 22. f(x)=-x + 1, h(x)=f(2x) 23. f(x)=-2x - 2, h(x)=f(5x) 24. f(x)=4x + 8, h(x)=\frac{1}{4}f(x)

Explanation:

Step1: Recall transformation rules

For a function $y = f(x)$, $y=-f(x)$ is a reflection over the $x - axis$, $y = f(-x)$ is a reflection over the $y - axis$, $y = af(x)$ with $|a|>1$ is a vertical stretch and $0<|a|<1$ is a vertical shrink, $y = f(bx)$ with $|b|>1$ is a horizontal shrink and $0<|b|<1$ is a horizontal stretch.

Step2: Analyze each exercise

Exercise 9

Given $h(x)=-f(x)$. By the rule, the graph of $h(x)$ is a reflection of the graph of $f(x)$ over the $x - axis$.

Exercise 10

Given $h(x)=f(-x)$. The graph of $h(x)$ is a reflection of the graph of $f(x)$ over the $y - axis$.

Exercise 11

Given $h(x)=f(-x)$. The graph of $h(x)$ is a reflection of the graph of $f(x)$ over the $y - axis$.

Exercise 12

Given $h(x)=-f(x)$. The graph of $h(x)$ is a reflection of the graph of $f(x)$ over the $x - axis$.

Exercise 13

Given $r(x) = 2f(x)$. Since $a = 2>1$, the graph of $r(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 2.

Exercise 14

Given $r(x)=f(\frac{1}{4}x)$. Since $b=\frac{1}{4}$ and $0 < b<1$, the graph of $r(x)$ is a horizontal stretch of the graph of $f(x)$ by a factor of 4.

Exercise 15

Given $r(x)=f(\frac{1}{2}x)$. Since $b = \frac{1}{2}$ and $0 < b<1$, the graph of $r(x)$ is a horizontal stretch of the graph of $f(x)$ by a factor of 2.

Exercise 16

Given $r(x)=f(\frac{1}{3}x)$. Since $b=\frac{1}{3}$ and $0 < b<1$, the graph of $r(x)$ is a horizontal stretch of the graph of $f(x)$ by a factor of 3.

Exercise 17

Given $r(x)=3f(x)$. Since $a = 3>1$, the graph of $r(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 3.

Exercise 18

Given $r(x)=4f(x)$. Since $a = 4>1$, the graph of $r(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 4.

Exercise 19

Given $h(x)=f(3x)$. Since $b = 3>1$, the graph of $h(x)$ is a horizontal shrink of the graph of $f(x)$ by a factor of $\frac{1}{3}$.

Exercise 20

Given $h(x)=\frac{1}{3}f(x)$. Since $a=\frac{1}{3}$ and $0 < a<1$, the graph of $h(x)$ is a vertical shrink of the graph of $f(x)$ by a factor of $\frac{1}{3}$.

Exercise 21

Given $h(x)=\frac{1}{6}f(x)$. Since $a=\frac{1}{6}$ and $0 < a<1$, the graph of $h(x)$ is a vertical shrink of the graph of $f(x)$ by a factor of $\frac{1}{6}$.

Exercise 22

Given $h(x)=f(2x)$. Since $b = 2>1$, the graph of $h(x)$ is a horizontal shrink of the graph of $f(x)$ by a factor of $\frac{1}{2}$.

Exercise 23

Given $h(x)=f(5x)$. Since $b = 5>1$, the graph of $h(x)$ is a horizontal shrink of the graph of $f(x)$ by a factor of $\frac{1}{5}$.

Exercise 24

Given $h(x)=\frac{3}{4}f(x)$. Since $a=\frac{3}{4}$ and $0 < a<1$, the graph of $h(x)$ is a vertical shrink of the graph of $f(x)$ by a factor of $\frac{3}{4}$.

Answer:

• Exercise 9: Reflection over the x - axis
• Exercise 10: Reflection over the y - axis
• Exercise 11: Reflection over the y - axis
• Exercise 12: Reflection over the x - axis
• Exercise 13: Vertical stretch by a factor of 2
• Exercise 14: Horizontal stretch by a factor of 4
• Exercise 15: Horizontal stretch by a factor of 2
• Exercise 16: Horizontal stretch by a factor of 3
• Exercise 17: Vertical stretch by a factor of 3
• Exercise 18: Vertical stretch by a factor of 4
• Exercise 19: Horizontal shrink by a factor of $\frac{1}{3}$
• Exercise 20: Vertical shrink by a factor of $\frac{1}{3}$
• Exercise 21: Vertical shrink by a factor of $\frac{1}{6}$
• Exercise 22: Horizontal shrink by a factor of $\frac{1}{2}$
• Exercise 23: Horizontal shrink by a factor of $\frac{1}{5}$
• Exercise 24: Vertical shrink by a factor of $\frac{3}{4}$