Question
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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 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 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
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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 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 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
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Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which equation in vertex form matches the graph of the firework trajectory, we need to analyze the characteristics of a parabola in vertex form, which is given by:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola, and \( a \) determines the direction (upward if \( a > 0 \) and downward if \( a < 0 \)) and the width of the parabola.
Since the parabola shown opens downward, the value of \( a \) must be negative.
Reviewing the provided options:
1. **y = 12(x − 5)² + 10** → Opens upwards (a positive value: 12)
2. **y = (1/2)(x + 5)² + 10** → Opens upwards (a positive value: 1/2)
3. **y = −12(x − 5)² + 10** → Opens downwards (a negative value: -12)
4. **y = −(1/2)(x − 5)² − 10** → Opens downwards (a negative value: -1/2)
Since the graph opens downward, we look for options 3 and 4.
Next, we need to identify the vertex. If the vertex appears to be at the point (5, 10) based on your description, then option 3 is the most appropriate because it maintains the vertex at \( (5, 10) \) (since it follows the form):
\[ y = -12(x - 5)^2 + 10 \]
Thus, the final answer is:
**y = -12(x − 5)² + 10**.
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola, and \( a \) determines the direction (upward if \( a > 0 \) and downward if \( a < 0 \)) and the width of the parabola.
Since the parabola shown opens downward, the value of \( a \) must be negative.
Reviewing the provided options:
1. **y = 12(x − 5)² + 10** → Opens upwards (a positive value: 12)
2. **y = (1/2)(x + 5)² + 10** → Opens upwards (a positive value: 1/2)
3. **y = −12(x − 5)² + 10** → Opens downwards (a negative value: -12)
4. **y = −(1/2)(x − 5)² − 10** → Opens downwards (a negative value: -1/2)
Since the graph opens downward, we look for options 3 and 4.
Next, we need to identify the vertex. If the vertex appears to be at the point (5, 10) based on your description, then option 3 is the most appropriate because it maintains the vertex at \( (5, 10) \) (since it follows the form):
\[ y = -12(x - 5)^2 + 10 \]
Thus, the final answer is:
**y = -12(x − 5)² + 10**.
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