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To solve Task 5, we need to find the x- and y-intercepts of the function \( f(x) = -2|x - 1| + 4 \) algebraically, and then sketch a rough graph with the provided intercepts, domain, and range.

### Step 1: Find the Y-Intercept
The y-intercept occurs when \( x = 0 \).

1. Substitute \( x = 0 \) into the function:
   \[
   f(0) = -2|0 - 1| + 4
   \]
2. Simplify the expression:
   \[
   f(0) = -2 \cdot 1 + 4 = -2 + 4 = 2
   \]
So, the y-intercept is \( (0, 2) \).

### Step 2: Find the X-Intercepts
The x-intercepts occur when \( f(x) = 0 \).

1. Set \( f(x) = 0 \):
   \[
   -2|x - 1| + 4 = 0
   \]
2. Rearrange to solve for \( |x - 1| \):
   \[
   -2|x - 1| = -4
   \]
   \[
   |x - 1| = 2
   \]
3. Solve \( |x - 1| = 2 \) by considering both cases:
   - Case 1: \( x - 1 = 2 \)
     \[
     x = 3
     \]
   - Case 2: \( x - 1 = -2 \)
     \[
     x = -1
     \]

So, the x-intercepts are \( (-1, 0) \) and \( (3, 0) \).

### Summary of Intercepts
- **Y-intercept**: \( (0, 2) \)
- **X-intercepts**: \( (-1, 0) \) and \( (3, 0) \)

### Additional Instructions for the Graph
Based on the instructions:
- **X-intercept** points should be \( (-2, 0) \) and \( (3, 0) \), though we found \( (-1, 0) \). Please confirm if this needs adjustment.
- **Y-intercept**: \( (0, 4) \), whereas we calculated \( (0, 2) \).
  
Would you like further clarification on these intercept differences or a more detailed explanation of the steps?

### Related Questions
1. How can we determine the range of a function involving absolute values?
2. What is the effect of a coefficient on an absolute value function, like \(-2\) in this case?
3. How do transformations affect the intercepts of absolute value functions?
4. What is the process for finding intercepts of functions with other types of expressions, like quadratic functions?
5. Why might the calculated intercepts differ from those given in the task instructions?

### Tip
When working with absolute value functions, it helps to separate cases for \( x \) when solving equations involving \( |x| \).

Identify the x and y-intercepts of f(x) = -2|x - 1| + 4 algebraically. Sketch a rough graph with X-intercept: (-2, 0), (3, 0); Y-intercept: (0, 4); Domain: (-4, 4); Range: [-4, 6].

Learn how to solve for the x- and y-intercepts of the function f(x) = -2|x - 1| + 4 algebraically, and understand how to sketch its graph. This guide covers intercepts, domain, and range for absolute value functions. Suitable for students in Grades 9-12.

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