Question
Identify the vertex, axis of symmetry, and direction of the graph of each function. Compare the width of the graph to the width of the graph of f(x)equals=x squaredx2.
f(x)equals=8 left parenthesis x plus 3 right parenthesis squared plus 28(x+3)2+2
Question content area bottom
Part 1
The vertex is enter your response here.
(Type an ordered
f(x)equals=8 left parenthesis x plus 3 right parenthesis squared plus 28(x+3)2+2
Question content area bottom
Part 1
The vertex is enter your response here.
(Type an ordered
Answers
GPT-4o mini
To find the vertex, axis of symmetry, and direction of the graph of the function \( f(x) = 8(x + 3)^2 + 2 \), we can analyze the given quadratic function.
### Standard Form
The function is already in vertex form, which is given by:
\[
f(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola and \( a \) determines the direction and width of the graph.
### Identifying Vertex
From the given function:
\[
f(x) = 8(x + 3)^2 + 2
\]
We can rewrite it in the standard form:
\[
f(x) = 8(x - (-3))^2 + 2
\]
Here, we can see that \( h = -3 \) and \( k = 2 \). Therefore, the vertex is:
\[
\text{Vertex} = (-3, 2)
\]
### Axis of Symmetry
The axis of symmetry for the parabola can be found using the \( h \) value from the vertex. It is given by:
\[
x = h
\]
Thus, the axis of symmetry is:
\[
x = -3
\]
### Direction of the Graph
Since \( a = 8 \) (which is positive), the parabola opens upwards.
### Width Comparison
To compare the width of the graph of this function to the width of the graph of \( f(x) = x^2 \):
1. The function \( f(x) = x^2 \) has a width of 1 because the coefficient (leading coefficient) is 1.
2. In \( f(x) = 8(x + 3)^2 + 2 \), the coefficient is 8. The width of the parabola is inversely proportional to the coefficient \( a \). Specifically, the width of a parabola can be calculated as \( \frac{1}{|a|} \).
Since \( a = 8 \):
- For \( f(x) = 8(x + 3)^2 + 2\), the width is \( \frac{1}{8} \).
- For \( f(x) = x^2\), the width is \( 1 \).
### Conclusion
- **Vertex**: (-3, 2)
- **Axis of Symmetry**: \( x = -3 \)
- **Direction**: Upward
- **Width Comparison**: The graph of \( f(x) = 8(x + 3)^2 + 2 \) is narrower than the graph of \( f(x) = x^2 \), as it is \( \frac{1}{8} \) compared to \( 1 \).
In summary, your answers are:
- **Vertex**: (-3, 2)
- **Axis of symmetry**: \( x = -3 \)
- **Direction of opening**: Upward
### Standard Form
The function is already in vertex form, which is given by:
\[
f(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola and \( a \) determines the direction and width of the graph.
### Identifying Vertex
From the given function:
\[
f(x) = 8(x + 3)^2 + 2
\]
We can rewrite it in the standard form:
\[
f(x) = 8(x - (-3))^2 + 2
\]
Here, we can see that \( h = -3 \) and \( k = 2 \). Therefore, the vertex is:
\[
\text{Vertex} = (-3, 2)
\]
### Axis of Symmetry
The axis of symmetry for the parabola can be found using the \( h \) value from the vertex. It is given by:
\[
x = h
\]
Thus, the axis of symmetry is:
\[
x = -3
\]
### Direction of the Graph
Since \( a = 8 \) (which is positive), the parabola opens upwards.
### Width Comparison
To compare the width of the graph of this function to the width of the graph of \( f(x) = x^2 \):
1. The function \( f(x) = x^2 \) has a width of 1 because the coefficient (leading coefficient) is 1.
2. In \( f(x) = 8(x + 3)^2 + 2 \), the coefficient is 8. The width of the parabola is inversely proportional to the coefficient \( a \). Specifically, the width of a parabola can be calculated as \( \frac{1}{|a|} \).
Since \( a = 8 \):
- For \( f(x) = 8(x + 3)^2 + 2\), the width is \( \frac{1}{8} \).
- For \( f(x) = x^2\), the width is \( 1 \).
### Conclusion
- **Vertex**: (-3, 2)
- **Axis of Symmetry**: \( x = -3 \)
- **Direction**: Upward
- **Width Comparison**: The graph of \( f(x) = 8(x + 3)^2 + 2 \) is narrower than the graph of \( f(x) = x^2 \), as it is \( \frac{1}{8} \) compared to \( 1 \).
In summary, your answers are:
- **Vertex**: (-3, 2)
- **Axis of symmetry**: \( x = -3 \)
- **Direction of opening**: Upward