Select ALL the correct answers. Consider the graph of function g below. A diagonal curve declines from (negative 3, 8), (negative 2, 5), (negative 1, 2), (0, negative 1), (1, negative 4), (2, negative 7), and (3, negative 10) on an x y coordinate plan
To determine which sequences of transformations could be applied to the parent function \( f(x) = x \) in order to obtain the graph of the function \( g \) based on the description provided, we need to analyze the characteristics of the function \( g \).
The described function has a decreasing pattern going from positive y-values to negative y-values:
- Starts from (−3, 8), moving down to (3, −10).
It appears that the graph of \( g \) is a downward-opening curve that starts at a relatively high point and moves downwards through the x-axis into negative y-values. This indicates that we might be dealing with a reflection and some vertical transformations.
### Let's analyze the transformations provided:
1. **Reflect over the x-axis, vertically stretch by a factor of 3, and then shift down 1 unit.**
- Reflecting over the x-axis changes the direction of the graph.
- Vertically stretching by a factor of 3 would enhance its steepness.
- Shifting down 1 unit maintains the reflection.
- **This transformation could indeed produce a similar curve as described.**
2. **Shift right 1 unit, reflect over the y-axis, and then vertically stretch by a factor of 3.**
- Shifting right would usually move the curve from the left to the right side of the graph.
- Reflecting over the y-axis changes its direction incorrectly based on the original position.
- This sequence does not match the characteristics given.
- **This transformation would not match.**
3. **Shift right 1 unit, reflect over the x-axis, and then vertically stretch by a factor of 3.**
- Shifting right would not align with the leftward points of the original.
- Reflecting over the x-axis will also misalign the curve.
- **This transformation is not consistent.**
4. **Reflect over the y-axis, vertically stretch by a factor of 3, and then shift down 1 unit.**
- Reflecting over the y-axis again creates a mismatch in the trajectory of the graph.
- This transformation also does not align with the start at the left part of the y-axis.
- **This transformation is not correct.**
5. **Shift left 2 units, reflect over the x-axis, and then vertically stretch by a factor of 3.**
- Shifting left can align the graph closer to starting up at positive y-values.
- Reflecting over the x-axis aligns the declining trend correctly.
- Finally stretching will enhance the steepness as needed.
- **This transformation could work.**
6. **Shift down 1 unit, reflect over the x-axis, and then vertically stretch by a factor of 3.**
- Shifting down would not align the starting positive points correctly with the given points.
- **This transformation is incorrect.**
### Therefore, the correct options are:
1. **Reflect over the x-axis, vertically stretch by a factor of 3, and then shift down 1 unit.**
5. **Shift left 2 units, reflect over the x-axis, and then vertically stretch by a factor of 3.**
### Final Answers:
- Reflect over the x-axis, vertically stretch by a factor of 3, and then shift down 1 unit.
- Shift left 2 units, reflect over the x-axis, and then vertically stretch by a factor of 3.