Question

the black graph is y = f(x). choose the equation for the red graph. y/2 = f(x) 2y = f(x) y = f(2x) y = f(x/2)

Explanation:

Step1: Recall horizontal - stretch/shrink rule

For a function $y = f(x)$, if we have $y=f(bx)$ where $b>1$, the graph of $y = f(x)$ is horizontally shrunk by a factor of $\frac{1}{b}$, and if $0 < b<1$, the graph of $y = f(x)$ is horizontally stretched by a factor of $\frac{1}{b}$.

Step2: Analyze the points on the graphs

The point $(2,4)$ on the black - graph corresponds to the point $(1,4)$ on the red - graph. The $x$ - coordinate of the point on the red - graph is half of the $x$ - coordinate of the corresponding point on the black - graph. This indicates a horizontal shrink by a factor of $\frac{1}{2}$.

Step3: Determine the transformation formula

The transformation from $y = f(x)$ to the new function for the red - graph is of the form $y = f(bx)$. Since the horizontal shrink factor is $\frac{1}{2}$, we have $b = 2$, so the equation of the red - graph is $y=f(2x)$.

Answer:

$y = f(2x)$