Question
What transformations produce the graph ofg left parenthesis x right parenthesis equals 5 superscript negative x plus 2 baseline from the graph of the parent function f left parenthesis x right parenthesis equals 5 superscript x baseline? Select all that apply.
(1 point)
Responses
reflection over the x-axis
reflection over the x -axis
reflection over the y-axis
reflection over the y -axis
horizontal shift to the left 2 units
horizontal shift to the left 2 units
horizontal shift to the right 2 units
horizontal shift to the right 2 units
vertical shift up 2 units
vertical shift up 2 units
vertical shift down 2 units
vertical shift down 2 units
Answers
GPT-4o mini
To describe the transformations that produce the graph of \( g(x) = 5^{-x} + 2 \) from the parent function \( f(x) = 5^x \), let's analyze the transformations step by step:
1. **Reflection over the x-axis**:
The function \( g(x) \) can be rewritten as \( g(x) = \frac{1}{5^x} + 2 \). The term \( 5^{-x} \) indicates a reflection of \( f(x) \) over the y-axis, but we'll be careful here. A negative exponent reflects the graph horizontally; therefore, this transformation does **not** involve reflection over the x-axis.
2. **Reflection over the y-axis**:
The transformation from \( f(x) = 5^x \) to \( g(x) = 5^{-x} \) is indeed a reflection over the y-axis.
3. **Horizontal shift to the left 2 units**:
There is no horizontal shift in this transformation.
4. **Horizontal shift to the right 2 units**:
There is no horizontal shift in this transformation.
5. **Vertical shift up 2 units**:
The \( +2 \) in \( g(x) = 5^{-x} + 2 \) indicates a vertical shift upward by 2 units.
6. **Vertical shift down 2 units**:
There is no vertical shift downward in this transformation.
### Therefore, the applicable transformations are:
- Reflection over the y-axis
- Vertical shift up 2 units
This means that the correct responses are:
- Reflection over the y-axis
- Vertical shift up 2 units
1. **Reflection over the x-axis**:
The function \( g(x) \) can be rewritten as \( g(x) = \frac{1}{5^x} + 2 \). The term \( 5^{-x} \) indicates a reflection of \( f(x) \) over the y-axis, but we'll be careful here. A negative exponent reflects the graph horizontally; therefore, this transformation does **not** involve reflection over the x-axis.
2. **Reflection over the y-axis**:
The transformation from \( f(x) = 5^x \) to \( g(x) = 5^{-x} \) is indeed a reflection over the y-axis.
3. **Horizontal shift to the left 2 units**:
There is no horizontal shift in this transformation.
4. **Horizontal shift to the right 2 units**:
There is no horizontal shift in this transformation.
5. **Vertical shift up 2 units**:
The \( +2 \) in \( g(x) = 5^{-x} + 2 \) indicates a vertical shift upward by 2 units.
6. **Vertical shift down 2 units**:
There is no vertical shift downward in this transformation.
### Therefore, the applicable transformations are:
- Reflection over the y-axis
- Vertical shift up 2 units
This means that the correct responses are:
- Reflection over the y-axis
- Vertical shift up 2 units