Question

in the graph below, the red graph is the parent function $y = \sqrt{x}$. the black graph is a dilation of $k$, $y = k\sqrt{x}$ where $k = $ _.

Explanation:

Step1: Find a point on the black graph

Looking at the black graph, let's pick a point. When \( x = 4 \), let's see the \( y \)-value. From the graph, when \( x = 4 \), \( y=-4 \) (since it's a dilation and the direction is downward, so negative).

Step2: Substitute into the equation \( y = k\sqrt{x} \)

We have the equation \( y=k\sqrt{x} \). Substitute \( x = 4 \) and \( y=-4 \) into it. So, \( -4=k\sqrt{4} \).

Step3: Solve for \( k \)

Since \( \sqrt{4}=2 \), the equation becomes \( -4 = k\times2 \). Then, divide both sides by 2: \( k=\frac{-4}{2}=-2 \).

Answer:

\( -2 \)