Asked by julian celeste
Use the image to answer the question.
A coordinate plane shows a parabola that opens downward. The x-axis ranges from 0 to 15 in increments of 5. The y-axis ranges from 0 to 15 in increments of 5.
The graph shows the trajectory of a firework. Which equation in vertex form matches this graph?
(1 point)
Responses
y=−12(x−5)2+10
y equals negative Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared plus 10
y=12(x+5)2+10
y equals Start Fraction 1 over 2 End Fraction left parenthesis x plus 5 right parenthesis squared plus 10
y=−12(x−5)2−10
y equals negative Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared minus 10
y=12(x−5)2+10
A coordinate plane shows a parabola that opens downward. The x-axis ranges from 0 to 15 in increments of 5. The y-axis ranges from 0 to 15 in increments of 5.
The graph shows the trajectory of a firework. Which equation in vertex form matches this graph?
(1 point)
Responses
y=−12(x−5)2+10
y equals negative Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared plus 10
y=12(x+5)2+10
y equals Start Fraction 1 over 2 End Fraction left parenthesis x plus 5 right parenthesis squared plus 10
y=−12(x−5)2−10
y equals negative Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared minus 10
y=12(x−5)2+10
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which equation in vertex form matches the given graph of a parabola that opens downward, we need to analyze the parameters of vertex form, which is:
\[ y = a(x - h)^2 + k \]
In this equation:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the direction and width of the parabola: if \( a < 0 \), the parabola opens downward, and if \( a > 0 \), it opens upward.
Based on your description:
- The parabola opens downward, indicating \( a < 0 \).
- The vertex appears to be at \( (5, 10) \).
Now, let's evaluate the options provided:
1. **\( y = -12(x - 5)^2 + 10 \)**:
- This is a downward opening parabola with vertex \( (5, 10) \) (correct form).
2. **\( y = \frac{-1}{2}(x - 5)^2 + 10 \)**:
- This is also a downward opening parabola with vertex \( (5, 10) \) (correct form but different \( a \)).
3. **\( y = -12(x - 5)^2 - 10 \)**:
- This is a downward opening parabola, but the vertex is at \( (5, -10) \) (incorrect vertex).
4. **\( y = 12(x - 5)^2 + 10 \)**:
- This is an upward opening parabola (incorrect direction).
Given this analysis, the equations that correctly feature a downward-opening parabola with the vertex at \( (5, 10) \) are the first two options.
If you are only looking for one correct response, **the best match would be:**
**\( y = -12(x - 5)^2 + 10 \)**
This option maintains the downward trajectory and has the correct vertex.
\[ y = a(x - h)^2 + k \]
In this equation:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the direction and width of the parabola: if \( a < 0 \), the parabola opens downward, and if \( a > 0 \), it opens upward.
Based on your description:
- The parabola opens downward, indicating \( a < 0 \).
- The vertex appears to be at \( (5, 10) \).
Now, let's evaluate the options provided:
1. **\( y = -12(x - 5)^2 + 10 \)**:
- This is a downward opening parabola with vertex \( (5, 10) \) (correct form).
2. **\( y = \frac{-1}{2}(x - 5)^2 + 10 \)**:
- This is also a downward opening parabola with vertex \( (5, 10) \) (correct form but different \( a \)).
3. **\( y = -12(x - 5)^2 - 10 \)**:
- This is a downward opening parabola, but the vertex is at \( (5, -10) \) (incorrect vertex).
4. **\( y = 12(x - 5)^2 + 10 \)**:
- This is an upward opening parabola (incorrect direction).
Given this analysis, the equations that correctly feature a downward-opening parabola with the vertex at \( (5, 10) \) are the first two options.
If you are only looking for one correct response, **the best match would be:**
**\( y = -12(x - 5)^2 + 10 \)**
This option maintains the downward trajectory and has the correct vertex.
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