Q. 34 - Precalculus Enhanced with Graphing Utilities

Question

If $\left(4,2\right)$ is a point on the graph of $y=f\left(x\right)$, which of the following points must be on the graph of $y=f2\left(x\right)$?

Step-by-Step Solution

Answer

The point$(2,2)$ lies on the graph of$y=f\left(2x\right)$ so the option c is correct 

Step 1. Given data

The point$\left(4,2\right)$ lies on the graph of the function$y=f\left(x\right)$

transformed function is$y=f\left(2x\right)$

Step 2. Explanation of correct option C

Point$(4,2)$ lies on $y=f\left(x\right)$

so $f\left(4\right)=2$

$y=f\left(x\right)$ stretched horizontally for $y=f\left(2x\right)$

so y coordinate of $y=f\left(2x\right)$at $x=2$

$$\begin{gathered} y=f\left(2x\right) \\ y=f(2\times 2) \\ y=f\left(4\right) \\ y=2 \end{gathered}$$

point$(2,2)$ lies on $y=f\left(2x\right)$so option b is correct.

Step 3. explanation of incorrect options

• x coordinate of$y=f\left(2x\right)$ is half of x coordinate of $y=f\left(x\right)$ so $(4,1)$ cannot lie on $y=f\left(2x\right)$
  
• x coordinate of $y=f\left(2x\right)$is half of x coordinate of $y=f\left(x\right)$ so$(8,2)$  cannot lie on $y=f\left(2x\right)$
• x coordinate of $y=f\left(2x\right)$is half of x coordinate of $y=f\left(x\right)$ so$(4,4)$ cannot lie on $y=f\left(2x\right)$