To solve this problem, we need to analyze the equation $y = |2x - 5|$ and answer the following questions:

1. Graph the equation by selecting the correct graph from the options.
2. Find the x-intercept.
3. Determine the intervals where the graph is increasing and decreasing.

Step-by-Step Solution:

Step 1: Analyze the Equation

The equation given is $y = |2x - 5|$. This is an absolute value function, which creates a V-shaped graph.

Step 2: Find the Vertex and X-Intercept

1. Vertex: The expression inside the absolute value, $2x - 5$, equals zero when $x = \frac{5}{2} = 2.5$. So, the vertex is at $\left(2.5, 0\right)$.
2. X-Intercept: Since the vertex is where the function touches the x-axis, the x-intercept is at $\left(2.5, 0\right)$.

Step 3: Determine Increasing and Decreasing Intervals

1. For $x < 2.5$: The slope of $2x - 5$ is negative (since it’s decreasing on this interval), so the function will be decreasing.
2. For $x > 2.5$: The slope of $2x - 5$ is positive, making the function increase.

Thus:

• Increasing on $(2.5, \infty)$
• Decreasing on $(-\infty, 2.5)$

Step 4: Match with the Graph

Now, based on these characteristics, we should look for a V-shaped graph with a vertex at $(2.5, 0)$, decreasing on the left and increasing on the right. Looking at the options:

• Option C has the correct vertex and interval characteristics.

Final Answers

• Correct Graph: Option C
• X-Intercept: $(2.5, 0)$
• Increasing Interval: $(2.5, \infty)$
• Decreasing Interval: $(-\infty, 2.5)$

Would you like further details on each step, or do you have any specific questions?

Here are five additional questions related to this topic:

1. How do you determine the vertex of an absolute value function?
2. What changes in the graph if the absolute value term is multiplied by a negative sign?
3. How would the graph change if we had $y = |2x + 3|$ instead?
4. Can you describe how to find intercepts for any absolute value function?
5. What are the differences in intervals if the slope within the absolute value changes?

Tip: For absolute value functions, the vertex is located where the expression inside the absolute value equals zero.