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Question
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A coordinate plane's x-axis ranges from negative 10 to 2 and its y-axis ranges from negative 4 to 4, both by 1-unit increments. A concave down parabola is drawn passing through the 2nd, 3rd, and 4th quadrants.
Which response is the correct way to write the equation in vertex form for the function?
(1 point)
Responses
y=−14(x+4)2+2
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x plus 4 right parenthesis squared plus 2
y=−14(x−2)2+4
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x minus 2 right parenthesis squared plus 4
y=−14(x−4)2+2
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x minus 4 right parenthesis squared plus 2
y=−14(x+2)2+4
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x plus 2 right parenthesis squared plus 4
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Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 10 to 2 and its y-axis ranges from negative 4 to 4, both by 1-unit increments. A concave down parabola is drawn passing through the 2nd, 3rd, and 4th quadrants.
Which response is the correct way to write the equation in vertex form for the function?
(1 point)
Responses
y=−14(x+4)2+2
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x plus 4 right parenthesis squared plus 2
y=−14(x−2)2+4
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x minus 2 right parenthesis squared plus 4
y=−14(x−4)2+2
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x minus 4 right parenthesis squared plus 2
y=−14(x+2)2+4
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x plus 2 right parenthesis squared plus 4
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Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine the correct vertex form of the parabola, we can analyze the given options and the characteristics described.
1. **Vertex Form of a Parabola**: The vertex form of a quadratic function is generally expressed as:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
2. **Concavity**: Since the parabola is described as concave down, we know that the coefficient \(a\) will be negative.
3. **Determining the Vertex from the Options**: We must choose the vertex that aligns with the characteristics of the graph observed in the coordinate plane described.
4. **Candidates Analysis**: To find the vertex based on the provided options:
- **Option 1**: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
- **Option 2**: \(y = -\frac{1}{4}(x - 2)^2 + 4\)
- **Option 3**: \(y = -\frac{1}{4}(x - 4)^2 + 2\)
- **Option 4**: \(y = -\frac{1}{4}(x + 2)^2 + 4\)
5. **Identify the Vertex**:
- For **Option 1**, vertex is at \((-4, 2)\).
- For **Option 2**, vertex is at \((2, 4)\).
- For **Option 3**, vertex is at \((4, 2)\).
- For **Option 4**, vertex is at \((-2, 4)\).
6. **Choose the Correct Option Based on Parabola's Location**:
Since the parabola is said to pass through 2nd, 3rd, and 4th quadrants, the vertex must be located between the quadrants effectively along the y-axis. Considering the y-coordinates and which x-coordinates still allow it to arch downwards while intersecting the defined quadrants, we find:
- **Quadrants**:
- 2nd quadrant: where \(x < 0\) and \(y > 0\).
- 3rd quadrant: where \(x < 0\) and \(y < 0\).
- 4th quadrant: where \(x > 0\) and \(y < 0\).
Considering the choices, Option 1 has a vertex positioned into quadrant 2 with a maximum height of 2 and opens downward, thereby intersecting the 3rd quadrant successfully.
Thus the correct answer is:
**Option 1**: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
1. **Vertex Form of a Parabola**: The vertex form of a quadratic function is generally expressed as:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
2. **Concavity**: Since the parabola is described as concave down, we know that the coefficient \(a\) will be negative.
3. **Determining the Vertex from the Options**: We must choose the vertex that aligns with the characteristics of the graph observed in the coordinate plane described.
4. **Candidates Analysis**: To find the vertex based on the provided options:
- **Option 1**: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
- **Option 2**: \(y = -\frac{1}{4}(x - 2)^2 + 4\)
- **Option 3**: \(y = -\frac{1}{4}(x - 4)^2 + 2\)
- **Option 4**: \(y = -\frac{1}{4}(x + 2)^2 + 4\)
5. **Identify the Vertex**:
- For **Option 1**, vertex is at \((-4, 2)\).
- For **Option 2**, vertex is at \((2, 4)\).
- For **Option 3**, vertex is at \((4, 2)\).
- For **Option 4**, vertex is at \((-2, 4)\).
6. **Choose the Correct Option Based on Parabola's Location**:
Since the parabola is said to pass through 2nd, 3rd, and 4th quadrants, the vertex must be located between the quadrants effectively along the y-axis. Considering the y-coordinates and which x-coordinates still allow it to arch downwards while intersecting the defined quadrants, we find:
- **Quadrants**:
- 2nd quadrant: where \(x < 0\) and \(y > 0\).
- 3rd quadrant: where \(x < 0\) and \(y < 0\).
- 4th quadrant: where \(x > 0\) and \(y < 0\).
Considering the choices, Option 1 has a vertex positioned into quadrant 2 with a maximum height of 2 and opens downward, thereby intersecting the 3rd quadrant successfully.
Thus the correct answer is:
**Option 1**: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
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