Describe the transformation of f(x)= x^2 represented by g. Then graph each function for g(x)=(1/4x)^2

### Transformation of \( f(x) = x^2 \) to \( g(x) = \left(\frac{1}{4}x\right)^2 \):

1. **Horizontal Stretch/Compression**:
   - The function \( g(x) = \left(\frac{1}{4}x\right)^2 \) represents a transformation of the parent function \( f(x) = x^2 \).
   - Specifically, the term \(\frac{1}{4}x\) indicates a **horizontal stretch** by a factor of 4. This is because the coefficient inside the function is \(\frac{1}{4}\), meaning each \( x \)-value of \( f(x) = x^2 \) is multiplied by 4 before squaring.

2. **Comparison**:
   - The graph of \( g(x) \) will be wider than the graph of \( f(x) = x^2 \) because stretching horizontally by a factor of 4 makes the parabola open more gradually.

### Graphing \( g(x) = \left(\frac{1}{4}x\right)^2 \):

1. **Key Points**:
   - Calculate the values of \( g(x) \) for key points of \( x \) to graph the function.
   - For example:
     - When \( x = 0 \), \( g(0) = 0 \).
     - When \( x = 4 \), \( g(4) = \left(\frac{1}{4} \times 4\right)^2 = 1 \).
     - When \( x = -4 \), \( g(-4) = \left(\frac{1}{4} \times -4\right)^2 = 1 \).
     - When \( x = 8 \), \( g(8) = \left(\frac{1}{4} \times 8\right)^2 = 4 \).
     - When \( x = -8 \), \( g(-8) = \left(\frac{1}{4} \times -8\right)^2 = 4 \).

2. **Shape**:
   - Plot these points and sketch the parabola, noting that it will be much wider compared to \( f(x) = x^2 \).

Would you like me to generate a graph for better visualization?

### Related Questions:
1. What does a horizontal stretch/compression do to the graph of a function?
2. How can you determine the factor of stretching or compressing a graph?
3. What is the impact of changing coefficients inside versus outside of function terms?
4. How would a vertical stretch affect the graph of \( f(x) = x^2 \)?
5. How does the transformation differ if the coefficient is outside the square, such as \( g(x) = \frac{1}{4}x^2 \)?

**Tip:** Coefficients inside the function affect the x-values, causing horizontal changes, while those outside affect y-values, causing vertical changes!

Explore the transformation of the quadratic function f(x) = x^2 to g(x) = (1/4x)^2. Learn about horizontal stretching/compression and graphing techniques. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation