The given equation is $y = |2x - 3|$.

(a) Find the x-intercept:

To find the x-intercept, set $y = 0$ and solve for $x$: $0 = |2x - 3|$ The absolute value equation $|2x - 3| = 0$ is true when: $2x - 3 = 0 \implies x = \frac{3}{2}$

So, the x-intercept is $\left(\frac{3}{2}, 0\right)$.

(b) Determine where the graph is increasing and where it is decreasing:

The graph of an absolute value function like $y = |2x - 3|$ forms a V-shape. It has a vertex where the expression inside the absolute value equals zero: $2x - 3 = 0 \implies x = \frac{3}{2}$

• Increasing: For $x > \frac{3}{2}$, the function increases because the slope of $2x - 3$ is positive.
• Decreasing: For $x < \frac{3}{2}$, the function decreases as the slope is negative on this side of the vertex.

Graph Identification:

Given the V-shape nature of the graph and the vertex at $x = \frac{3}{2}$, the correct graph should show a V with its vertex at $\left(\frac{3}{2}, 0\right)$. From the provided options (A, B, C, D), we need to select the graph that matches this description.

Based on the x-intercept and the shape of the graph around $x = \frac{3}{2}$, the correct answer is Option D.

Would you like further details or have any questions about this solution?

Related Questions:

1. What is the vertex form of an absolute value function?
2. How do absolute value functions change when you alter the coefficients?
3. How do you find the y-intercept of $y = |2x - 3|$?
4. What is the general form of the graph of an absolute value function?
5. How does the graph of $y = |2x - 3|$ compare to $y = 2x - 3$?

Tip:

For absolute value functions of the form $y = |ax + b|$, the graph always has a V-shape with the vertex at the point where $ax + b = 0$.