Question
\begin{aligned}
f(x)&=x^2\\\\
g(x)&=-\dfrac{1}{2}x^2
\end{aligned}
graph these
f(x)&=x^2\\\\
g(x)&=-\dfrac{1}{2}x^2
\end{aligned}
graph these
Answers
Answered by
GPT-4o mini
To graph the functions \( f(x) = x^2 \) and \( g(x) = -\frac{1}{2}x^2 \), follow these steps:
1. **Identify the type of functions**:
- \( f(x) = x^2 \) is a standard parabola that opens upwards with its vertex at the origin (0,0).
- \( g(x) = -\frac{1}{2}x^2 \) is also a parabola but it opens downwards and is wider than the standard parabola because the coefficient of \( x^2 \) is less than -1.
2. **Plot key points**:
- For \( f(x) = x^2 \):
- When \( x = -2 \), \( f(-2) = 4 \)
- When \( x = -1 \), \( f(-1) = 1 \)
- When \( x = 0 \), \( f(0) = 0 \)
- When \( x = 1 \), \( f(1) = 1 \)
- When \( x = 2 \), \( f(2) = 4 \)
- For \( g(x) = -\frac{1}{2}x^2 \):
- When \( x = -2 \), \( g(-2) = -2 \)
- When \( x = -1 \), \( g(-1) = -0.5 \)
- When \( x = 0 \), \( g(0) = 0 \)
- When \( x = 1 \), \( g(1) = -0.5 \)
- When \( x = 2 \), \( g(2) = -2 \)
3. **Draw the axes**:
- Create a coordinate system with \( x \) and \( y \) axes.
4. **Plot the points**:
- For \( f(x) \), plot the points: (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4).
- For \( g(x) \), plot the points: (-2, -2), (-1, -0.5), (0, 0), (1, -0.5), (2, -2).
5. **Sketch the curves**:
- Connect the points for \( f(x) \) with a smooth curve to form an upward-opening parabola.
- Connect the points for \( g(x) \) with a smooth curve to form a downward-opening parabola.
### Graph Overview
- **Y-axis**: The \( f(x) = x^2 \) parabola is upward, symmetric around the y-axis, reaching higher values for both positive and negative \( x \).
- **Y-axis**: The \( g(x) = -\frac{1}{2}x^2 \) parabola is downward, also symmetric around the y-axis, but it reaches lower values as \( |x| \) increases.
### Visual Representation
If you want an actual plot, you would typically use graphing software or tools like Desmos, GeoGebra, or a graphing calculator to create the visual representation of these functions. But based on the coordinates and the behavior discussed, you can sketch the graphs on paper.
1. **Identify the type of functions**:
- \( f(x) = x^2 \) is a standard parabola that opens upwards with its vertex at the origin (0,0).
- \( g(x) = -\frac{1}{2}x^2 \) is also a parabola but it opens downwards and is wider than the standard parabola because the coefficient of \( x^2 \) is less than -1.
2. **Plot key points**:
- For \( f(x) = x^2 \):
- When \( x = -2 \), \( f(-2) = 4 \)
- When \( x = -1 \), \( f(-1) = 1 \)
- When \( x = 0 \), \( f(0) = 0 \)
- When \( x = 1 \), \( f(1) = 1 \)
- When \( x = 2 \), \( f(2) = 4 \)
- For \( g(x) = -\frac{1}{2}x^2 \):
- When \( x = -2 \), \( g(-2) = -2 \)
- When \( x = -1 \), \( g(-1) = -0.5 \)
- When \( x = 0 \), \( g(0) = 0 \)
- When \( x = 1 \), \( g(1) = -0.5 \)
- When \( x = 2 \), \( g(2) = -2 \)
3. **Draw the axes**:
- Create a coordinate system with \( x \) and \( y \) axes.
4. **Plot the points**:
- For \( f(x) \), plot the points: (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4).
- For \( g(x) \), plot the points: (-2, -2), (-1, -0.5), (0, 0), (1, -0.5), (2, -2).
5. **Sketch the curves**:
- Connect the points for \( f(x) \) with a smooth curve to form an upward-opening parabola.
- Connect the points for \( g(x) \) with a smooth curve to form a downward-opening parabola.
### Graph Overview
- **Y-axis**: The \( f(x) = x^2 \) parabola is upward, symmetric around the y-axis, reaching higher values for both positive and negative \( x \).
- **Y-axis**: The \( g(x) = -\frac{1}{2}x^2 \) parabola is downward, also symmetric around the y-axis, but it reaches lower values as \( |x| \) increases.
### Visual Representation
If you want an actual plot, you would typically use graphing software or tools like Desmos, GeoGebra, or a graphing calculator to create the visual representation of these functions. But based on the coordinates and the behavior discussed, you can sketch the graphs on paper.
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