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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.

If the given equations are graphed in the xy-plane, at how many points do the graphs of the equations intersect?
1. y = 18
2. y = -3(x - 18)^2 + 15

Learn how to solve a system of equations involving a horizontal line and a parabola. This algebra problem covers key concepts in graphing quadratic equations and finding intersection points. Suitable for Grades 9-12.

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