Question

function g is defined as $g(x) = \frac{1}{2}f(x)$. what is the graph of g? choose 1 answer: options a, b, c, d with their respective graphs

Explanation:

Step1: Understand the transformation

The function \( g(x)=\frac{1}{2}f(x) \) represents a vertical compression of the graph of \( f(x) \) by a factor of \( \frac{1}{2} \). This means every \( y \)-coordinate of the points on \( f(x) \) will be multiplied by \( \frac{1}{2} \).

Step2: Analyze key points of \( f(x) \)

Looking at the graph of \( f(x) \):

• The point at \( x = 0 \) has \( y = 2 \), so for \( g(x) \), \( y=\frac{1}{2}\times2 = 1 \)? Wait, no, wait the original \( f(x) \) at \( x = 0 \) is \( y = 2 \)? Wait no, looking at the blue graph, at \( x = 0 \), \( y = 2 \); at \( x = 2 \), \( y=-2 \); at \( x=-4 \) to \( x = 0 \), the \( y \)-value is 2. Wait, no, let's re - check. Wait the blue graph: the horizontal segment is at \( y = 2 \) from \( x=-4 \) to \( x = 0 \), then a line down to \( (2,-2) \), then up to \( (4,0) \).

So key points of \( f(x) \):

• For the horizontal part: \( y = 2 \) (from \( x=-4 \) to \( x = 0 \))
• At \( x = 2 \), \( y=-2 \)
• At \( x = 4 \), \( y = 0 \)

Step3: Apply the transformation to key points

• For the horizontal segment (\( y = 2 \)): After transformation, \( y=\frac{1}{2}\times2=1 \)
• At \( x = 2 \), \( y=\frac{1}{2}\times(-2)=-1 \)
• At \( x = 4 \), \( y=\frac{1}{2}\times0 = 0 \)
• At \( x=-4 \), \( y=\frac{1}{2}\times2 = 1 \)

Now let's check the options:

• Option A: The \( y \)-values are too large (e.g., at \( x = 0 \), \( y = 4 \)), which is a vertical stretch, not compression.
• Option B: The \( y \)-values are not compressed correctly (e.g., at \( x = 2 \), \( y = 0 \) which is wrong, should be \( y=-1 \))
• Option C: Let's check the points. The horizontal segment has \( y = 1 \) (since \( 2\times\frac{1}{2}=1 \)), at \( x = 2 \), \( y=-1 \) (since \( -2\times\frac{1}{2}=-1 \)), and at \( x = 4 \), \( y = 0 \) (since \( 0\times\frac{1}{2}=0 \)). This matches the vertical compression by \( \frac{1}{2} \).
• Option D: The shape and values do not match the compression.

Answer:

C