What transformations produce the graph of g(x) = 5^-x+2 from the graph of the parent function f(x) =5^x? Select all that apply. (1 point)
reflection over the x-axis
reflection over the y-axis
horizontal shift to the left 2 units
horizontal shift to the right 2 units
vertical shift up 2 units
vertical shift down 2 units
The correct transformations that produce the graph of g(x) = 5^(-x+2) from the graph of f(x) = 5^x are:
- Horizontal shift to the right 2 units
- Vertical shift up 2 units
To determine the transformations that produce the graph of g(x) = 5^(-x + 2) from the graph of the parent function f(x) = 5^x, we will analyze each option:
1. Reflection over the x-axis: This transformation would change the signs of the y-coordinates and reflect the graph vertically. It is not applied in the given function.
2. Reflection over the y-axis: This transformation would change the signs of the x-coordinates and reflect the graph horizontally. It is not applied in the given function.
3. Horizontal shift to the left 2 units: The expression -x + 2 inside the exponent represents a horizontal shift to the right 2 units. Therefore, the correct transformation is a horizontal shift to the right 2 units.
4. Horizontal shift to the right 2 units: This is the correct transformation, as explained above.
5. Vertical shift up 2 units: The +2 outside the exponent indicates a vertical shift up 2 units. This transformation is applied in the given function.
6. Vertical shift down 2 units: The function does not include a vertical shift down 2 units.
Therefore, the transformations that produce the graph of g(x) = 5^(-x + 2) from the graph of the parent function f(x) = 5^x are:
- Horizontal shift to the right 2 units
- Vertical shift up 2 units
Select all that apply:
- Horizontal shift to the right 2 units
- Vertical shift up 2 units
To determine the transformations that produce the graph of g(x) = 5^(-x+2) from the graph of f(x) = 5^x, we need to understand the effects of each transformation on the parent function.
1. Reflection over the x-axis:
This transformation would occur if we multiply the entire function by -1. However, in this case, g(x) = 5^(-x+2) is already positive for all values of x. So, there is no reflection over the x-axis.
2. Reflection over the y-axis:
This transformation would occur if we replace x with -x in the function. Since g(x) = 5^(-x+2), there is no reflection over the y-axis.
3. Horizontal shift to the left 2 units:
To shift a graph to the left, we subtract the specified value from x within the function. In this case, to shift f(x) = 5^x to the left by 2 units, we would have g(x) = 5^(-x-2).
4. Horizontal shift to the right 2 units:
To shift a graph to the right, we add the specified value to x within the function. However, in this case, we want to shift to the left, not the right. So, there is no horizontal shift to the right by 2 units.
5. Vertical shift up 2 units:
To shift a graph vertically, we add or subtract the specified value from the function. In this case, to shift f(x) = 5^x upward by 2 units, we would have g(x) = 5^x + 2.
6. Vertical shift down 2 units:
To shift a graph vertically, we add or subtract the specified value from the function. In this case, to shift f(x) = 5^x downward by 2 units, we would have g(x) = 5^x - 2.
From the above analysis, the transformations that produce the graph of g(x) = 5^(-x+2) from the graph of f(x) = 5^x are:
- Horizontal shift to the left 2 units: g(x) = 5^(-x-2)
- Vertical shift up 2 units: g(x) = 5^x + 2
- Vertical shift down 2 units: g(x) = 5^x - 2