The problem involves two functions:

• A linear function $f(x) = 5x - 13$,
• A quadratic function $g(x) = ax^2 + bx + c$.

The graphs of these functions intersect at points $A$ and $B$. The task is to find the $y$-coordinate of point $B$.

Let me analyze the problem in detail.

Step 1: Find the x-coordinates of the intersection points

The two functions intersect when $f(x) = g(x)$, so: $5x - 13 = ax^2 + bx + c$

Rearrange: $ax^2 + (b - 5)x + (c + 13) = 0$

This is a quadratic equation in $x$. From the graph:

• Point $A$ has $x \approx 3$,
• Point $B$ has $x \approx -1$.

Thus, the roots of the quadratic equation are approximately $x = 3$ and $x = -1$.

Using this information, the quadratic equation can be written as: $a(x - 3)(x + 1) = 0$

Expanding: $ax^2 - 2ax - 3a = 0$

Comparing coefficients with $ax^2 + (b - 5)x + (c + 13) = 0$, we can determine $b$ and $c$ in terms of $a$.

Step 2: Calculate the y-coordinate of point B

Point $B$ has $x = -1$. Substituting $x = -1$ into $f(x)$ gives the $y$-coordinate for point $B$: $f(-1) = 5(-1) - 13 = -5 - 13 = -18$

Thus, the $y$-coordinate of point $B$ is $-18$.

Let me know if you want further clarification or details.

Related Questions:

1. How do we derive the quadratic equation from two points of intersection?
2. What is the value of the coefficient $a$ in the quadratic function $g(x)$?
3. How do we confirm that $x = -1$ satisfies both functions?
4. Can we derive the full equation for $g(x)$ using the graph?
5. How is the vertex of the parabola $g(x)$ related to the intersection points?

Tip:

When working with intersections of functions, equating the functions and finding the roots of the resulting equation is essential.