Question
Use the drawing tools to form the correct answer on the graph.
Draw a line for the axis of symmetry of function f. Also mark the x-intercept(s), y-intercept, and vertex of the function.
f(x) = -(x + 1)^2+ 4
Draw a line for the axis of symmetry of function f. Also mark the x-intercept(s), y-intercept, and vertex of the function.
f(x) = -(x + 1)^2+ 4
Answers
GPT-4o mini
To analyze the quadratic function \( f(x) = -(x + 1)^2 + 4 \), we can identify several key features:
1. **Vertex**: The vertex form of a parabola is given by \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. In this case, the function is written as:
\[
f(x) = -1(x + 1)^2 + 4
\]
This means \( h = -1 \) and \( k = 4 \). Therefore, the vertex of the function is \( (-1, 4) \).
2. **Axis of Symmetry**: The axis of symmetry for a quadratic function \( f(x) = a(x - h)^2 + k \) is the vertical line passing through the vertex. For our function, the axis of symmetry is:
\[
x = -1
\]
3. **Y-intercept**: To find the y-intercept, we set \( x = 0 \) in the function:
\[
f(0) = - (0 + 1)^2 + 4 = -1 + 4 = 3
\]
Thus, the y-intercept is \( (0, 3) \).
4. **X-intercepts**: To find the x-intercepts, we set \( f(x) = 0 \):
\[
0 = - (x + 1)^2 + 4
\]
Rearranging gives:
\[
(x + 1)^2 = 4
\]
Taking square roots results in two solutions:
\[
x + 1 = 2 \quad \text{or} \quad x + 1 = -2
\]
This gives us:
\[
x = 1 \quad \text{or} \quad x = -3
\]
Therefore, the x-intercepts are \( (1, 0) \) and \( (-3, 0) \).
5. **Summary of Key Points**:
- Vertex: \( (-1, 4) \)
- Axis of symmetry: \( x = -1 \)
- X-intercepts: \( (1, 0) \) and \( (-3, 0) \)
- Y-intercept: \( (0, 3) \)
To visually represent this on the graph:
- Draw a vertical line at \( x = -1 \) for the axis of symmetry.
- Plot the vertex point at \( (-1, 4) \).
- Mark the y-intercept at point \( (0, 3) \).
- Mark the x-intercepts at points \( (1, 0) \) and \( (-3, 0) \).
If you require a sketch or drawing based on these instructions, please follow these guidelines to form the complete graph accurately.
1. **Vertex**: The vertex form of a parabola is given by \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. In this case, the function is written as:
\[
f(x) = -1(x + 1)^2 + 4
\]
This means \( h = -1 \) and \( k = 4 \). Therefore, the vertex of the function is \( (-1, 4) \).
2. **Axis of Symmetry**: The axis of symmetry for a quadratic function \( f(x) = a(x - h)^2 + k \) is the vertical line passing through the vertex. For our function, the axis of symmetry is:
\[
x = -1
\]
3. **Y-intercept**: To find the y-intercept, we set \( x = 0 \) in the function:
\[
f(0) = - (0 + 1)^2 + 4 = -1 + 4 = 3
\]
Thus, the y-intercept is \( (0, 3) \).
4. **X-intercepts**: To find the x-intercepts, we set \( f(x) = 0 \):
\[
0 = - (x + 1)^2 + 4
\]
Rearranging gives:
\[
(x + 1)^2 = 4
\]
Taking square roots results in two solutions:
\[
x + 1 = 2 \quad \text{or} \quad x + 1 = -2
\]
This gives us:
\[
x = 1 \quad \text{or} \quad x = -3
\]
Therefore, the x-intercepts are \( (1, 0) \) and \( (-3, 0) \).
5. **Summary of Key Points**:
- Vertex: \( (-1, 4) \)
- Axis of symmetry: \( x = -1 \)
- X-intercepts: \( (1, 0) \) and \( (-3, 0) \)
- Y-intercept: \( (0, 3) \)
To visually represent this on the graph:
- Draw a vertical line at \( x = -1 \) for the axis of symmetry.
- Plot the vertex point at \( (-1, 4) \).
- Mark the y-intercept at point \( (0, 3) \).
- Mark the x-intercepts at points \( (1, 0) \) and \( (-3, 0) \).
If you require a sketch or drawing based on these instructions, please follow these guidelines to form the complete graph accurately.