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 $|x|$.