Question

Use the graph of y equals f left parenthesis x right parenthesis to graph the function g left parenthesis x right parenthesis equals negative f left parenthesis x right parenthesis minus 1.
y = f(x)
-6
-4
-2
2
4
6
-6
-4
-2
2
4
6
x
y



A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A series of connected line segments have the following endpoints: (negative 2, 2); (0, 2); (2, 2).
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.
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Question content area right
Part 1
Choose the correct graph of g below.
A.
-6
6
-6
6
x
y



A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A series of connected line segments have the following endpoints: (negative 2, negative 1); (0, negative 1); (2, negative 1).
B.
-6
6
-6
6
x
y



A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A series of connected line segments have the following endpoints: (negative 2, 3); (0, 3); (2, 3).
C.
-6
6
-6
6
x
y



A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A series of connected line segments have the following endpoints: (negative 2, negative 3); (0, negative 3); (2, negative 3).
D.
-6
6
-6
6
x
y

Answer 1

To graph the function \( g(x) = -f(x) - 1 \), we will analyze how the transformations affect the original function \( f(x) \).

1. **Inverse Reflection**: The term \(-f(x)\) indicates that the graph of \( f(x) \) is reflected across the x-axis. This means that all the y-values of \( f(x) \) will become their negatives.

2. **Vertical Shift**: The term \(-1\) indicates that the entire graph of \(-f(x)\) will be shifted down by 1 unit.

Now, let's suppose the original function has the following characteristics based on the description provided:

- The key points of \( f(x) \) are:
- \( f(-2) = 2 \)
- \( f(0) = 2 \)
- \( f(2) = 2 \)

Calculating \( g(x) \) at those points:
- \( g(-2) = -f(-2) - 1 = -2 - 1 = -3 \)
- \( g(0) = -f(0) - 1 = -2 - 1 = -3 \)
- \( g(2) = -f(2) - 1 = -2 - 1 = -3 \)

This means \( g(x) \) will have the same x-values, but the y-values will now all be \(-3\).

Based on these transformations, the graph of \( g(x) \) will have the following endpoints:
- \( (-2, -3) \)
- \( (0, -3) \)
- \( (2, -3) \)

The correct graph of \( g(x) \), which is a horizontal line at \( y = -3 \) passing through these x-values, would match the description in option **C**:

**C.**
- Contains points: \( (-2, -3) \), \( (0, -3) \), and \( (2, -3) \), which is what we derived.

So the answer is **C**.