Question
Show me a graph using the function Y equals X raised to the second power -4
Answers
GPT-4o mini
I'm unable to create visual graphs directly, but I can describe how you can graph the function \( Y = X^2 - 4 \) and what the graph looks like.
### Steps to Graph \( Y = X^2 - 4 \):
1. **Identify the Type of Function**: This is a quadratic function, and its graph will be a parabola.
2. **Find the Vertex**: The vertex form of a quadratic function is \( Y = a(X - h)^2 + k \). In this case, it can be expressed as \( Y = 1(X - 0)^2 - 4 \), so the vertex is at the point \( (0, -4) \).
3. **Determine the Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is \( X = 0 \).
4. **Calculate Additional Points**: You can pick values for \( X \) and calculate \( Y \):
- If \( X = -2 \), then \( Y = (-2)^2 - 4 = 4 - 4 = 0 \) (point (-2, 0))
- If \( X = -1 \), then \( Y = (-1)^2 - 4 = 1 - 4 = -3 \) (point (-1, -3))
- If \( X = 1 \), then \( Y = (1)^2 - 4 = 1 - 4 = -3 \) (point (1, -3))
- If \( X = 2 \), then \( Y = (2)^2 - 4 = 4 - 4 = 0 \) (point (2, 0))
5. **Sketch the Graph**:
- Plot the vertex \( (0, -4) \).
- Plot the additional points: (-2, 0), (-1, -3), (1, -3), (2, 0).
- Draw a smooth curve through these points, forming a parabola that opens upwards.
### Features of the Graph:
- The graph opens upwards because the coefficient of \( X^2 \) is positive.
- The y-intercept is at \( (0, -4) \), and the x-intercepts (where the graph crosses the x-axis) are at \( (-2, 0) \) and \( (2, 0) \).
This should give you a good idea of how to visualize and sketch the graph of the function \( Y = X^2 - 4 \). If you have graphing software or a graphing calculator, you can also input the function directly to visualize it.
### Steps to Graph \( Y = X^2 - 4 \):
1. **Identify the Type of Function**: This is a quadratic function, and its graph will be a parabola.
2. **Find the Vertex**: The vertex form of a quadratic function is \( Y = a(X - h)^2 + k \). In this case, it can be expressed as \( Y = 1(X - 0)^2 - 4 \), so the vertex is at the point \( (0, -4) \).
3. **Determine the Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is \( X = 0 \).
4. **Calculate Additional Points**: You can pick values for \( X \) and calculate \( Y \):
- If \( X = -2 \), then \( Y = (-2)^2 - 4 = 4 - 4 = 0 \) (point (-2, 0))
- If \( X = -1 \), then \( Y = (-1)^2 - 4 = 1 - 4 = -3 \) (point (-1, -3))
- If \( X = 1 \), then \( Y = (1)^2 - 4 = 1 - 4 = -3 \) (point (1, -3))
- If \( X = 2 \), then \( Y = (2)^2 - 4 = 4 - 4 = 0 \) (point (2, 0))
5. **Sketch the Graph**:
- Plot the vertex \( (0, -4) \).
- Plot the additional points: (-2, 0), (-1, -3), (1, -3), (2, 0).
- Draw a smooth curve through these points, forming a parabola that opens upwards.
### Features of the Graph:
- The graph opens upwards because the coefficient of \( X^2 \) is positive.
- The y-intercept is at \( (0, -4) \), and the x-intercepts (where the graph crosses the x-axis) are at \( (-2, 0) \) and \( (2, 0) \).
This should give you a good idea of how to visualize and sketch the graph of the function \( Y = X^2 - 4 \). If you have graphing software or a graphing calculator, you can also input the function directly to visualize it.