Question
Which of the following graphs show a function of the form f(x)=(x-c)^2 +3 for some real number c? Select two that apply.
Answers
Answered by
GPT-4o mini
To determine which graphs represent the function of the form \( f(x) = (x-c)^2 + 3 \) for some real number \( c \), we need to recognize the characteristics of this quadratic function.
1. **Vertex**: The graph of \( f(x) = (x-c)^2 + 3 \) is a parabola that opens upwards. The vertex of this parabola is at the point \( (c, 3) \).
2. **Minimum Value**: The minimum value of the function occurs at the vertex, which is \( 3 \). This means the graph should touch or be at \( y = 3 \) for some value of \( x \).
3. **Shape**: Since it is a quadratic function opening upwards, all other points on the graph must be above or equal to \( y = 3 \).
To select the appropriate graphs, look for:
- A parabola that opens upwards.
- A minimum height of \( y = 3 \) (the vertex may touch the line \( y = 3 \)).
- The vertex somewhere along the horizontal axis (x-axis) but at a height of \( y = 3 \).
If you provide specific options for the graphs, I can help determine which ones fit this criteria!
1. **Vertex**: The graph of \( f(x) = (x-c)^2 + 3 \) is a parabola that opens upwards. The vertex of this parabola is at the point \( (c, 3) \).
2. **Minimum Value**: The minimum value of the function occurs at the vertex, which is \( 3 \). This means the graph should touch or be at \( y = 3 \) for some value of \( x \).
3. **Shape**: Since it is a quadratic function opening upwards, all other points on the graph must be above or equal to \( y = 3 \).
To select the appropriate graphs, look for:
- A parabola that opens upwards.
- A minimum height of \( y = 3 \) (the vertex may touch the line \( y = 3 \)).
- The vertex somewhere along the horizontal axis (x-axis) but at a height of \( y = 3 \).
If you provide specific options for the graphs, I can help determine which ones fit this criteria!