We are given two equations:

1. $y = 18$
2. $y = -3(x - 18)^2 + 15$

We need to determine how many points the graphs of these equations intersect when plotted in the xy-plane.

Step 1: Analyze the first equation $y = 18$

This is a horizontal line at $y = 18$. It represents all points where the y-coordinate is 18, regardless of the x-coordinate.

Step 2: Analyze the second equation $y = -3(x - 18)^2 + 15$

This is a downward-opening parabola with its vertex at $(18, 15)$. The vertex is found by inspecting the equation in vertex form: $y = a(x - h)^2 + k$ Here, $h = 18$ and $k = 15$, so the vertex is $(18, 15)$. The parabola opens downward because the coefficient of the squared term is negative (specifically, $-3$).

Step 3: Solve for the intersection points

We set the two equations equal to each other to find the points of intersection: $18 = -3(x - 18)^2 + 15$ Simplifying: $18 - 15 = -3(x - 18)^2$ $3 = -3(x - 18)^2$ $(x - 18)^2 = -1$ This equation has no real solutions because the square of a real number cannot be negative. Therefore, there are no points where the two graphs intersect.

Conclusion:

The two graphs do not intersect at any points, so the answer is $\boxed{D}$, which is Zero.

Would you like further explanation on any step?

Here are 5 related questions to explore:

1. What is the general form of a quadratic equation, and how do you find its vertex?
2. How do you solve a system of equations graphically?
3. What does it mean for two lines or curves to have exactly one intersection point?
4. Can a line and a parabola intersect at more than two points?
5. What are the implications of a negative coefficient in front of the squared term in a parabola?

Tip: Always inspect the form of equations to predict their general shape before solving for intersections.