Question

Use the graph of y equals f left parenthesis x right parenthesis shown to the right to graph the following function g.
g left parenthesis x right parenthesis equals negative f left parenthesis one half x right parenthesis minus 2
-8
8
-2
8
x
y
left parenthesis negative 3 comma 1 right parenthesisleft parenthesis 3 comma 1 right parenthesisleft parenthesis 0 comma 1 right parenthesisy equals f left parenthesis x right parenthesis



A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 2 to 8 in increments of 1. A horizontal line segment labeled y=f(x) crosses the y-axis at the point labeled (0, 1). The line segment connects the labeled points (negative 3, 1) and (3, 1).
Question content area bottom
Part 1
Choose the correct graph below.
A.
-8
8
-8
8
x
y



A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. A horizontal line segment crosses the y-axis at the plotted point (0, negative 3). The line segment connects the plotted points (negative 6, negative 3) and (6, negative 3).
B.
-8
8
-8
8
x
y



A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. A horizontal line segment crosses the y-axis at the plotted point (0, 3). The line segment connects the plotted points (negative 5, 3) and (5, 3).
C.
-8
8
-8
8
x
y



A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. A horizontal line segment crosses the y-axis at the plotted point (0, negative 1). The line segment connects the plotted points (negative 3, negative 1) and (3, negative 1).
D.
-8
8
-8
8
x
y

Answer 1

To graph the function \( g(x) = -f\left(\frac{1}{2}x\right) - 2 \) based on the behavior of \( f(x) \), we need to analyze the transformations applied to \( f(x) \).

1. **Horizontal Compression/Stretch**:
The term \( \frac{1}{2}x \) in \( f\left(\frac{1}{2}x\right) \) indicates a horizontal stretch by a factor of 2. This means for any point \( (a, b) \) on the graph of \( f(x) \), the corresponding point on \( f\left(\frac{1}{2}x\right) \) will be \( (2a, b) \).

2. **Reflection and Vertical Shift**:
The negative sign in front of \( f \) indicates a reflection over the x-axis. Thus, if \( (x, y) \) is on the graph of \( f \), then \( (x, -y) \) will be on the graph of \( -f\left(\frac{1}{2}x\right) \).
The term \( -2 \) indicates a downward shift of 2 units. Therefore, every y-coordinate will decrease by 2.

### Summary of Transformations:

- For points on \( f(x) \):
- \( (x, y) \) becomes \( (2x, -y - 2) \)

### Steps to Identify the New Points:

1. Looking at the original \( f(x) \), we have the points:
- \( (-3, 1) \)
- \( (0, 1) \)
- \( (3, 1) \)

2. Applying Transformations:
- For \( (-3, 1) \):
- \( x = -3 \) becomes \( x = 2 \cdot (-3) = -6 \)
- \( y = 1 \) becomes \( y = -1 - 2 = -3 \)
- New point: \( (-6, -3) \)

- For \( (0, 1) \):
- \( x = 0 \) becomes \( x = 2 \cdot 0 = 0 \)
- \( y = 1 \) becomes \( y = -1 - 2 = -3 \)
- New point: \( (0, -3) \)

- For \( (3, 1) \):
- \( x = 3 \) becomes \( x = 2 \cdot 3 = 6 \)
- \( y = 1 \) becomes \( y = -1 - 2 = -3 \)
- New point: \( (6, -3) \)

### Final Points for \( g(x) \):
We have derived the transformed points:
- \( (-6, -3) \)
- \( (0, -3) \)
- \( (6, -3) \)
These points suggest that \( g(x) \) is a horizontal line at \( y = -3 \) connecting \( (-6, -3) \) and \( (6, -3) \).

### Choosing the Correct Graph:
Based on our analysis, we can find the graph that corresponds to a horizontal line at \( y = -3 \) which connects from \( (-6, -3) \) to \( (6, -3) \).

From the provided options:
- **Graph A** appears to be the correct choice because it has the horizontal line at \( y = -3 \) from \( x = -6 \) to \( x = 6 \).

Thus, the answer is **A**.