Engineers are designing a stretch of highway on a mountain. They plan to put a lookout station at the highest point called summit. The equation of the transitional parabola is y=-0.0000333xsquared +0.03x+120. Determine the coordinates at the summit.

The equation provided represents a parabola in the form:

\[
y = -0.0000333x^2 + 0.03x + 120
\]

This is a quadratic equation. To find the coordinates of the summit (the highest point of the parabola), we need to determine the vertex of the parabola. The formula to find the x-coordinate of the vertex for a quadratic equation \(y = ax^2 + bx + c\) is:

\[
x = \frac{-b}{2a}
\]

Here:
- \(a = -0.0000333\)
- \(b = 0.03\)

Let's calculate the x-coordinate of the vertex and then substitute it back into the equation to find the corresponding y-coordinate.The coordinates of the summit (the highest point of the parabola) are approximately:

\[
(450.45, 126.76)
\]

This means the lookout station should be placed at \(x = 450.45\) meters along the highway, and it will be at an elevation of \(126.76\) meters.

Would you like more details on how this was calculated, or do you have any other questions?

Here are 5 related questions to expand on this topic:
1. How does the shape of the parabola change with different coefficients of \(a\)?
2. What is the significance of the y-intercept in this context?
3. How would you find where the highway intersects a specific elevation?
4. How can you determine the total length of the highway based on this parabola?
5. How does the steepness of the slope affect the driving experience?

**Tip:** The vertex form of a quadratic equation is particularly useful when focusing on the maximum or minimum point of a parabola, such as in this scenario.

Explore how to find the coordinates of the summit on a parabolic highway using the vertex form of a quadratic equation. Learn step-by-step with an example equation y = -0.0000333x^2 + 0.03x + 120. Suitable for high school students studying quadratic equations and vertex calculations.

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