Question
Which of the graphs shown below represents the base function f(x)=x−−√
(in black) and the stretched function g(x)=−32x−−√
in red.
(in black) and the stretched function g(x)=−32x−−√
in red.
Answers
Answered by
GPT-4o mini
To visualize and compare the base function \( f(x) = \sqrt{x} \) and the stretched function \( g(x) = -\frac{3}{2} \sqrt{x} \), we need to consider their properties:
1. **Base Function \( f(x) = \sqrt{x} \)**:
- This function is defined for \( x \geq 0 \).
- It starts at the origin (0,0) and rises slowly, increasing as \( x \) increases. The graph is a curve that gets less steep as it moves to the right.
2. **Stretched Function \( g(x) = -\frac{3}{2} \sqrt{x} \)**:
- This function is also defined for \( x \geq 0 \).
- Unlike \( f(x) \), \( g(x) \) is negative, meaning it reflects the base function \( f(x) \) across the x-axis.
- The stretching factor of \(-\frac{3}{2}\) means that for every value of \( \sqrt{x} \), it is scaled down vertically by a factor of \( \frac{3}{2} \) and then reflected. This results in the graph starting at (0,0) but moving downwards as \( x \) increases.
To determine which graph represents these functions:
- Look for a graph where the black curve starts at the origin (0,0) and rises gently as \( x \) increases (this will be \( f(x) = \sqrt{x} \)).
- Then, look for a red curve that starts at the same point (0,0) but moves downward since it is negative and stretches downward.
You would identify \( f(x) = \sqrt{x} \) in black and \( g(x) = -\frac{3}{2}\sqrt{x} \) in red based on these characteristics. If you have visual graphs available, follow these traits to match each one correctly. If you need further assistance or explanation, feel free to ask!
1. **Base Function \( f(x) = \sqrt{x} \)**:
- This function is defined for \( x \geq 0 \).
- It starts at the origin (0,0) and rises slowly, increasing as \( x \) increases. The graph is a curve that gets less steep as it moves to the right.
2. **Stretched Function \( g(x) = -\frac{3}{2} \sqrt{x} \)**:
- This function is also defined for \( x \geq 0 \).
- Unlike \( f(x) \), \( g(x) \) is negative, meaning it reflects the base function \( f(x) \) across the x-axis.
- The stretching factor of \(-\frac{3}{2}\) means that for every value of \( \sqrt{x} \), it is scaled down vertically by a factor of \( \frac{3}{2} \) and then reflected. This results in the graph starting at (0,0) but moving downwards as \( x \) increases.
To determine which graph represents these functions:
- Look for a graph where the black curve starts at the origin (0,0) and rises gently as \( x \) increases (this will be \( f(x) = \sqrt{x} \)).
- Then, look for a red curve that starts at the same point (0,0) but moves downward since it is negative and stretches downward.
You would identify \( f(x) = \sqrt{x} \) in black and \( g(x) = -\frac{3}{2}\sqrt{x} \) in red based on these characteristics. If you have visual graphs available, follow these traits to match each one correctly. If you need further assistance or explanation, feel free to ask!