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We are given the equation of the quadratic function: 

\[
y = a(x - 4)^2 + 8
\]

We are also provided with a point on the parabola: \((4, 4)\), and we need to find the value of \(r\), which is the x-coordinate of the intersection point between the quadratic function and a linear function \(l\).

### Step 1: Use the point (4, 4) to find \(a\)
Substitute \(x = 4\) and \(y = 4\) into the quadratic equation to solve for \(a\):

\[
4 = a(4 - 4)^2 + 8
\]

This simplifies to:

\[
4 = a(0)^2 + 8
\]

\[
4 = 8
\]

This equation is clearly false, indicating an issue. Let's try to interpret the graph further.

The graph shows that the vertex of the parabola is at \((4, 8)\), not \((4, 4)\) as the provided point indicates. So the given point \((4, 4)\) is **not** the vertex, but simply a point on the parabola. Let's correct the approach and go ahead with solving the value of \(r\).

Would you like me to proceed with a correction and the rest of the process?

The xy-plane shows graphs of a linear function and a quadratic function y = a(x - 4)^2 + 8, where a is a constant. If (r, s) is the point of intersection, what is the value of r?

This problem explores the intersection between a quadratic function y = a(x - 4)^2 + 8 and a linear function. Using key algebraic techniques, learn how to find the x-coordinate of the intersection (r) by solving for a and applying systems of equations. Suitable for high school students.

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