Let's solve the problem step-by-step.

Problem Analysis

1. Given Equation: $y = |2x - 13|$
2. Tasks:
   • Find the x-intercept.
   • Determine intervals where the graph is increasing and decreasing.
   • Identify the correct graph from options A, B, C, and D.

Solution Steps

Step 1: Finding the X-Intercept

To find the x-intercept, set $y = 0$: $0 = |2x - 13|$ This implies: $2x - 13 = 0$ Solving for $x$: $x = \frac{13}{2} = 6.5$ So, the x-intercept is $(6.5, 0)$.

Step 2: Determine Intervals of Increase and Decrease

Since this is an absolute value function, it has a vertex (turning point) where the expression inside the absolute value equals zero: $2x - 13 = 0 \Rightarrow x = 6.5$

• For $x < 6.5$: $y = 13 - 2x$, which is a decreasing function as $x$ increases.
• For $x > 6.5$: $y = 2x - 13$, which is an increasing function as $x$ increases.

Thus:

• The graph is increasing on the interval $(6.5, \infty)$.
• The graph is decreasing on the interval $(-\infty, 6.5)$.

Step 3: Identify the Correct Graph

From these observations:

• The correct graph should have a vertex at $(6.5, 0)$, with the graph decreasing to the left of $x = 6.5$ and increasing to the right.
• Option A seems to match this description, with a V-shape centered at $(6.5, 0)$, decreasing on the left and increasing on the right.

Summary of Answers

• X-intercept: $(6.5, 0)$
• Increasing interval: $(6.5, \infty)$
• Decreasing interval: $(-\infty, 6.5)$
• Correct Graph: Option A

Would you like further clarification on any of these steps?

Related Questions

1. How do you find the x-intercept of an absolute value function in general?
2. Why does the absolute value function create a V-shaped graph?
3. What are other methods to determine increasing and decreasing intervals for more complex functions?
4. How would the solution change if the function were $y = |3x - 10|$ instead?
5. What transformations affect the shape and position of an absolute value graph?

Tip

Absolute value graphs form V-shapes because they mirror the linear part of the function across the x-axis when the expression inside becomes negative.