Q. 35 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that the x-intercepts of the graph of $y=f\left(x\right)$ are$-5$ and $3$.

(a) What are the x-intercepts of the graph of $y=f(x+2)$?

(b) What are the x-intercepts of the graph of $y=f(x-2)$?

(c) What are the x-intercepts of the graph of $y=4f\left(x\right)$?

(d) What are the x-intercepts of the graph of $y=f(-x)$?

Step-by-Step Solution

Answer

(a) x-intercepts of the graph of $y=f(x+2)$ are $-7 \& 1$

(b) x-intercepts of the graph of $y=f(x-2)$ are $-3 \& 5$

(c) x-intercepts of the graph of $y=4f\left(x\right)$ are $-5 \& 3$

(d) x-intercepts of the graph of $y=f(-x)$ are $5 \& -3$ 

Step 1. Given data

x-intercepts of $y=f\left(x\right)$ are $-5 \& 3$

Transformed functions are

(a) $y=f(x+2)$

(b) $y=f(x-2)$

(c) $y=4f\left(x\right)$

(d) $y=f(-x)$

Step 2. part a

Graph of function$y=f(x+2)$ can be obtained by shifting the graph of function $y=f\left(x\right)$ towards left by two units

x coordinates of all points of$y=f(x+2)$ will be two units less than the x coordinates of points of $y=f\left(x\right)$

so x-intercepts of $y=f(x+2)$ will be

$$\begin{gathered} -5-2=-7 \\ 3-2=1 \end{gathered}$$

Step 3. part b

Graph of function $y=f(x-2)$can be obtained by shifting the graph of function $y=f\left(x\right)$ towards the right by two units

x coordinates of all points of$y=f(x-2)$ will be two units greater than the x coordinates of points of $y=f\left(x\right)$

so x-intercepts of function$y=f(x-2)$ will be

$$\begin{gathered} -5+2=-3 \\ 3+2=5 \end{gathered}$$

Step 4. part c

Graph of function$y=4f\left(x\right)$ can be obtained by vertically stretching the graph of the function $y=f\left(x\right)$ by the factor of $4$

x-intercepts of both functions will be the same 

so x-intercepts of $y=4f\left(x\right)$ will be $-5\&3$

Step 5. part d

Graph of function $y=f(-x)$can be obtained reflecting graph of function$y=f\left(x\right)$ across the y axis 

x coordinates of all points of$y=f(-x)$ will be the same magnitude as the x coordinates of points of $y=f\left(x\right)$ but with an opposite sign

so x-intercepts of  will be$5 \& -3$