Question

The function f(x)=2(x+2^2) is shown in the graph. Which table shows that g(x)=−f(−x) results in a reflection about the origin of the function graphed? Hint: Make a table of the x - and y -coordinates based on the graph. Use these points to determine the coordinates of the function reflected about the origin. (1 point) Responses x g(x) 1 −2 2 0 3 −2 x f left parenthesis x right parenthesis 1 negative 2 2 0 3 negative 2 x g(x) −1 −2 −2 0 −3 −2 x f left parenthesis x right parenthesis negative 1 negative 2 negative 2 0 negative 3 negative 2 x g(x) −1 2 −2 0 −3 2 x f left parenthesis x right parenthesis negative 1 2 negative 2 0 negative 3 2 x g(x) 1 2 2 0 3 2 x f left parenthesis x right parenthesis 1 2 2 0 3 2 Skip to navigation

Answer 1

To determine which table shows that \( g(x) = -f(-x) \) results in a reflection about the origin of the function graphed, let's break down what \( g(x) \) represents in terms of the original function \( f(x) \).

1. Calculate \( -f(-x) \):
- For any point \( (x, f(x)) \), when we substitute \(-x\), we get \( f(-x) \).
- Applying the negative sign gives us \( -f(-x) \).
- A reflection about the origin transforms the point \( (x, f(x)) \) into \( (-x, -f(x)) \).

2. Therefore, for each point \( (x, f(x)) \) in the original function, the point \( g(x) = -f(-x) \) will correspond to \( (-x, -f(x)) \).

3. For a specific example, if we have points from the graph of \( f(x) \):
- Say, if \( f(1) = 2 \), then the reflected point would be \( g(-1) = -f(-1) \). If \( f(-1) = -2 \), then \( g(-1) = 2 \).

We need to verify each response against these transformations:

1. **Option 1** (g(x) table):
```
x g(x)
1 -2
2 0
3 -2
```

Corresponds to \( (1, -2), (2, 0), (3, -2) \). If we assume \( f(1)=2 \), \( f(2)=0 \), and \( f(3)=2 \), we see that it does not reflect points about the origin correctly.

2. **Option 2** (g(x) table):
```
x g(x)
-1 -2
-2 0
-3 -2
```

Corresponds to \( (-1, -2), (-2, 0), (-3, -2) \). Again, this does not correctly correspond to the origin reflection.

3. **Option 3** (g(x) table):
```
x g(x)
-1 2
-2 0
-3 2
```

This corresponds to \( (-1, 2), (-2, 0), (-3, 2) \) which can be verified if we take their original counterparts \( (1, -2), (2, 0), (3, -2) \) and reflect.

4. **Option 4** (g(x) table):
```
x g(x)
1 2
2 0
3 2
```

This corresponds to points reflecting up, which doesn't show a reflection about the origin.

Upon reviewing these options, *Option 3* is the correct one, as it reflects the points about the origin correctly based on the transformations given.

Therefore, the correct response is **Option 3**:
```
x g(x)
-1 2
-2 0
-3 2
```