Question
graph the function f left parenthesis x right parenthesis equals negative one half x squared
Answers
GPT-4o mini
To graph the function \( f(x) = -\frac{1}{2} x^2 \), which is a quadratic function opening downwards, you can follow these steps:
1. **Identify Key Features:**
- **Vertex:** Since the function is in the standard form \( ax^2 + bx + c \), the vertex is at the maximum point for a downward-opening parabola. Here, the vertex is at the origin \((0, 0)\).
- **Axis of Symmetry:** This is the vertical line \( x = 0 \).
- **Direction:** The parabola opens downwards because the coefficient of \( x^2 \) (which is \(-\frac{1}{2}\)) is negative.
2. **Calculate Some Points:**
- Choose some values for \( x \) and calculate \( f(x) \):
- \( f(0) = -\frac{1}{2}(0)^2 = 0 \)
- \( f(2) = -\frac{1}{2}(2)^2 = -2 \)
- \( f(-2) = -\frac{1}{2}(-2)^2 = -2 \)
- \( f(1) = -\frac{1}{2}(1)^2 = -\frac{1}{2} \)
- \( f(-1) = -\frac{1}{2}(-1)^2 = -\frac{1}{2} \)
3. **Plot Points:**
- You have the following points to plot:
- \( (0, 0) \)
- \( (2, -2) \)
- \( (-2, -2) \)
- \( (1, -\frac{1}{2}) \)
- \( (-1, -\frac{1}{2}) \)
4. **Sketch the Graph:**
- Start by plotting the points on a Cartesian plane.
- Draw a smooth curve through the points, ensuring that it opens downwards and is symmetric about the y-axis.
### Sketch:
Here’s a rough sketch of the graph of the function \( f(x) = -\frac{1}{2} x^2 \):
```
|
0.4| *
| *
0.2| *
| *
0.0|---*-------------*---*-----> x
| * *
-0.2| *
| *
-0.4| *
|
-1.0|
|
-1.5|
|
-2.0| * *
|_____________________________
-2 0 2
```
### Conclusion:
The graph of \( f(x) = -\frac{1}{2} x^2 \) is a downward-opening parabola with its vertex at the origin, and it decreases as you move away from the vertex along the x-axis.
1. **Identify Key Features:**
- **Vertex:** Since the function is in the standard form \( ax^2 + bx + c \), the vertex is at the maximum point for a downward-opening parabola. Here, the vertex is at the origin \((0, 0)\).
- **Axis of Symmetry:** This is the vertical line \( x = 0 \).
- **Direction:** The parabola opens downwards because the coefficient of \( x^2 \) (which is \(-\frac{1}{2}\)) is negative.
2. **Calculate Some Points:**
- Choose some values for \( x \) and calculate \( f(x) \):
- \( f(0) = -\frac{1}{2}(0)^2 = 0 \)
- \( f(2) = -\frac{1}{2}(2)^2 = -2 \)
- \( f(-2) = -\frac{1}{2}(-2)^2 = -2 \)
- \( f(1) = -\frac{1}{2}(1)^2 = -\frac{1}{2} \)
- \( f(-1) = -\frac{1}{2}(-1)^2 = -\frac{1}{2} \)
3. **Plot Points:**
- You have the following points to plot:
- \( (0, 0) \)
- \( (2, -2) \)
- \( (-2, -2) \)
- \( (1, -\frac{1}{2}) \)
- \( (-1, -\frac{1}{2}) \)
4. **Sketch the Graph:**
- Start by plotting the points on a Cartesian plane.
- Draw a smooth curve through the points, ensuring that it opens downwards and is symmetric about the y-axis.
### Sketch:
Here’s a rough sketch of the graph of the function \( f(x) = -\frac{1}{2} x^2 \):
```
|
0.4| *
| *
0.2| *
| *
0.0|---*-------------*---*-----> x
| * *
-0.2| *
| *
-0.4| *
|
-1.0|
|
-1.5|
|
-2.0| * *
|_____________________________
-2 0 2
```
### Conclusion:
The graph of \( f(x) = -\frac{1}{2} x^2 \) is a downward-opening parabola with its vertex at the origin, and it decreases as you move away from the vertex along the x-axis.