Question
The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
Question content area bottom
Part 1
Drag each function given above into the area below the appropriate graph, depending on which function is represented by which graph.
1.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens upward and has its vertex at (0, negative 25).
2.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens upward and has its vertex at (negative 5, 0).
3.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens upward and has its vertex at (5, 0).
4.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens downward and has its vertex at (0, negative 25).
Question content area bottom
Part 1
Drag each function given above into the area below the appropriate graph, depending on which function is represented by which graph.
1.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens upward and has its vertex at (0, negative 25).
2.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens upward and has its vertex at (negative 5, 0).
3.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens upward and has its vertex at (5, 0).
4.
-60
-30
30
60
-60
-30
30
60
x
y
A coordinate system has a horizontal x axis labeled from negative 60 to 60 in increments of 30 and a vertical y axis labeled from negative 60 to 60 in increments of 30. The xy plane contains a parabola that opens downward and has its vertex at (0, negative 25).
Answers
GPT-4o mini
To identify the equations of the quadratic functions based on the vertex and the direction in which the parabola opens, we can use the vertex form of a quadratic equation, which is:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
Now, let's analyze each of the graphs based on the provided vertex coordinates and whether the parabola opens upward or downward.
### Graph 1
- **Vertex**: (0, -25)
- **Direction**: Opens upward
Using the vertex form:
\[
y = a(x - 0)^2 - 25 \implies y = ax^2 - 25
\]
The value of \(a\) must be positive since the parabola opens upwards.
### Graph 2
- **Vertex**: (-5, 0)
- **Direction**: Opens upward
Using the vertex form:
\[
y = a(x + 5)^2 + 0 \implies y = a(x + 5)^2
\]
Again, since it opens upwards, \(a > 0\).
### Graph 3
- **Vertex**: (5, 0)
- **Direction**: Opens upward
Using the vertex form:
\[
y = a(x - 5)^2 + 0 \implies y = a(x - 5)^2
\]
Again, \(a > 0\).
### Graph 4
- **Vertex**: (0, -25)
- **Direction**: Opens downward
Using the vertex form:
\[
y = a(x - 0)^2 - 25 \implies y = -a(x^2) - 25
\]
Here, since it opens downwards, \(a < 0\).
### Summary of Function Options
- For **Graph 1**: Upward, vertex (0, -25) - likely a function with a positive coefficient for \(x^2\).
- For **Graph 2**: Upward, vertex (-5, 0) - likely a function with a positive coefficient for \((x + 5)^2\).
- For **Graph 3**: Upward, vertex (5, 0) - likely a function with a positive coefficient for \((x - 5)^2\).
- For **Graph 4**: Downward, vertex (0, -25) - likely a function with a negative coefficient for \(x^2\).
Based on this analysis, you would match each function to its corresponding graph by looking at the starting vertex and the direction of opening:
1. **Graph 1** corresponds to a quadratic function with vertex (0, -25) opening upward.
2. **Graph 2** corresponds to a quadratic function with vertex (-5, 0) opening upward.
3. **Graph 3** corresponds to a quadratic function with vertex (5, 0) opening upward.
4. **Graph 4** corresponds to a quadratic function with vertex (0, -25) opening downward.
Make sure to refer back to your list of provided function options to precisely match them to these derived equations.
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
Now, let's analyze each of the graphs based on the provided vertex coordinates and whether the parabola opens upward or downward.
### Graph 1
- **Vertex**: (0, -25)
- **Direction**: Opens upward
Using the vertex form:
\[
y = a(x - 0)^2 - 25 \implies y = ax^2 - 25
\]
The value of \(a\) must be positive since the parabola opens upwards.
### Graph 2
- **Vertex**: (-5, 0)
- **Direction**: Opens upward
Using the vertex form:
\[
y = a(x + 5)^2 + 0 \implies y = a(x + 5)^2
\]
Again, since it opens upwards, \(a > 0\).
### Graph 3
- **Vertex**: (5, 0)
- **Direction**: Opens upward
Using the vertex form:
\[
y = a(x - 5)^2 + 0 \implies y = a(x - 5)^2
\]
Again, \(a > 0\).
### Graph 4
- **Vertex**: (0, -25)
- **Direction**: Opens downward
Using the vertex form:
\[
y = a(x - 0)^2 - 25 \implies y = -a(x^2) - 25
\]
Here, since it opens downwards, \(a < 0\).
### Summary of Function Options
- For **Graph 1**: Upward, vertex (0, -25) - likely a function with a positive coefficient for \(x^2\).
- For **Graph 2**: Upward, vertex (-5, 0) - likely a function with a positive coefficient for \((x + 5)^2\).
- For **Graph 3**: Upward, vertex (5, 0) - likely a function with a positive coefficient for \((x - 5)^2\).
- For **Graph 4**: Downward, vertex (0, -25) - likely a function with a negative coefficient for \(x^2\).
Based on this analysis, you would match each function to its corresponding graph by looking at the starting vertex and the direction of opening:
1. **Graph 1** corresponds to a quadratic function with vertex (0, -25) opening upward.
2. **Graph 2** corresponds to a quadratic function with vertex (-5, 0) opening upward.
3. **Graph 3** corresponds to a quadratic function with vertex (5, 0) opening upward.
4. **Graph 4** corresponds to a quadratic function with vertex (0, -25) opening downward.
Make sure to refer back to your list of provided function options to precisely match them to these derived equations.