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In this problem, you are asked to complete the sketch of an even function \( y = g(x) \) for the domain \( -5 \leq x \leq 5 \), given that the sketch for \( y = 4x - x^2 \) is shown for \( 0 \leq x \leq 5 \).

### Step-by-step Solution:

1. **Even Function Properties**:  
   An even function satisfies the condition:
   \[
   g(x) = g(-x)
   \]
   This means that the graph of an even function is symmetric with respect to the y-axis.

2. **Given Function**:
   You are given the function:
   \[
   g(x) = 4x - x^2 \quad \text{for} \quad 0 \leq x \leq 5
   \]
   The graph of this quadratic function is already plotted for the positive x-values between 0 and 5.

3. **Sketch for Negative x-values**:
   To complete the graph for the range \( -5 \leq x \leq 0 \), you need to reflect the given portion over the y-axis. This reflection will create the mirror image of the graph from \( x = 0 \) to \( x = 5 \) for the interval \( x = -5 \) to \( x = 0 \).

### Procedure to Sketch:

- **Plot the symmetric points**:
  - At \( x = 0 \), the point \( (0, 0) \) is on the graph.
  - For \( x = 1 \), you have \( g(1) = 4(1) - (1)^2 = 3 \). Therefore, \( g(-1) = 3 \).
  - For \( x = 2 \), you have \( g(2) = 4(2) - (2)^2 = 8 - 4 = 4 \). Thus, \( g(-2) = 4 \).
  - Similarly, for all \( x \) values from 0 to 5, reflect the corresponding y-values for negative x-values.

- **Connect the points smoothly** to reflect the symmetry about the y-axis.

Would you like a detailed image explanation of this sketch, or any further clarifications?

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Here are 5 questions to consider for deeper understanding:

1. What are the properties of even functions and how do they affect the shape of the graph?
2. How does symmetry about the y-axis impact the reflection of a function?
3. Can odd functions be sketched similarly? Why or why not?
4. What is the maximum value of the function \( g(x) = 4x - x^2 \), and where does it occur?
5. What are the roots (zeroes) of the function \( g(x) = 4x - x^2 \)?

**Tip**: When sketching even functions, always check for symmetry with respect to the y-axis by evaluating the function at both positive and negative x-values.

If this function is part of an even function y = g(x) defined for -5 ≤ x ≤ 5, sketch y = g(x). The sketch on the right shows the function y = 4x - x^2 for 0 ≤ x ≤ 5.

Learn how to complete the sketch of an even function y = g(x) given the function y = 4x - x^2 for the range 0 ≤ x ≤ 5. Understand the properties of even functions and how symmetry about the y-axis affects their graph. Suitable for students in Grades 9-11.

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