Question
Quadratic functions p and q are graphed on the grid. The graph of p was transformed to create the graph of q.
A. q(x) = -x^2 +2
B. q(x) = -(x + 2)^2
C. q(x) = -(x+2)^2
D. 9(x) = -x^2 - 2
A. q(x) = -x^2 +2
B. q(x) = -(x + 2)^2
C. q(x) = -(x+2)^2
D. 9(x) = -x^2 - 2
Answers
GPT-4o mini
To determine which function represents the transformed graph of p to create the graph of q, we need to analyze the provided equations and their characteristics.
Let's break down the options given:
1. **A. q(x) = -x² + 2**
This is a downward-opening parabola with a vertex at (0, 2). It shows a vertical shift upward by 2 from the standard form.
2. **B. q(x) = -(x + 2)²**
This is also a downward-opening parabola, but the vertex is located at (-2, 0). This represents a horizontal shift to the left by 2 units compared to the parent function (-x²).
3. **C. q(x) = -(x + 2)²**
This option is identical to B and has a downward-opening parabola with a vertex at (-2, 0).
4. **D. q(x) = -x² - 2**
This function is a downward-opening parabola with a vertex located at (0, -2). It represents a vertical shift downwards by 2 units from the parent function.
To determine the correct transformation from p to q, we need to know what the original function p(x) looks like. If p(x) were a standard upward-opening parabola (like x²), then:
- If q(x) is transformed with a vertical reflection (i.e., from p(x) upwards to q(x) downwards), it must include a negative sign in front of it.
- If there is a horizontal shift as well, we should look for the (x + c) form in the equations.
- Vertical shifts will adjust the constant at the end.
If we assume p(x) = x², the transformations resulting in q(x) could include:
1. **Reflections**: Negative sign indicating it opens down.
2. **Horizontal shifts**: Indicates a shift left/right.
3. **Vertical shifts**: Raise or lower entire graph.
Given the options A through D, if we need clarity on the transformations that have occurred to go from p(x) to q(x), we need additional context on what p(x) looks like originally.
If we assume that p(x) = -x² (which is a simple transformation to negative), then q(x) could be:
- If p was reflected and shifted left by 2, we get B or C.
- If it was simply reflected down and then shifted vertically, we get A or D.
The functions B and C appear to be the closest, as they show a horizontal shift of 2 units to the left and retain the negative sign, leading to a downward shape.
So, without additional information about p(x), we cannot definitively determine which transformation occurred; however, B and C are likely candidates for transformations with horizontal shifts.
With B and C being identical, if you only need one answer, either can be an answer, say B as per choice repetition.
Ultimately, please verify with the original shape of p for an exact match with q.
Let's break down the options given:
1. **A. q(x) = -x² + 2**
This is a downward-opening parabola with a vertex at (0, 2). It shows a vertical shift upward by 2 from the standard form.
2. **B. q(x) = -(x + 2)²**
This is also a downward-opening parabola, but the vertex is located at (-2, 0). This represents a horizontal shift to the left by 2 units compared to the parent function (-x²).
3. **C. q(x) = -(x + 2)²**
This option is identical to B and has a downward-opening parabola with a vertex at (-2, 0).
4. **D. q(x) = -x² - 2**
This function is a downward-opening parabola with a vertex located at (0, -2). It represents a vertical shift downwards by 2 units from the parent function.
To determine the correct transformation from p to q, we need to know what the original function p(x) looks like. If p(x) were a standard upward-opening parabola (like x²), then:
- If q(x) is transformed with a vertical reflection (i.e., from p(x) upwards to q(x) downwards), it must include a negative sign in front of it.
- If there is a horizontal shift as well, we should look for the (x + c) form in the equations.
- Vertical shifts will adjust the constant at the end.
If we assume p(x) = x², the transformations resulting in q(x) could include:
1. **Reflections**: Negative sign indicating it opens down.
2. **Horizontal shifts**: Indicates a shift left/right.
3. **Vertical shifts**: Raise or lower entire graph.
Given the options A through D, if we need clarity on the transformations that have occurred to go from p(x) to q(x), we need additional context on what p(x) looks like originally.
If we assume that p(x) = -x² (which is a simple transformation to negative), then q(x) could be:
- If p was reflected and shifted left by 2, we get B or C.
- If it was simply reflected down and then shifted vertically, we get A or D.
The functions B and C appear to be the closest, as they show a horizontal shift of 2 units to the left and retain the negative sign, leading to a downward shape.
So, without additional information about p(x), we cannot definitively determine which transformation occurred; however, B and C are likely candidates for transformations with horizontal shifts.
With B and C being identical, if you only need one answer, either can be an answer, say B as per choice repetition.
Ultimately, please verify with the original shape of p for an exact match with q.