https://file.aiquickdraw.com/mathbot/file/1727403438770xnxpfo8a.jpg

The problem provided is about a projectile motion scenario, where Gregg hits a golf ball and the task is to model its path using a quadratic equation. Here’s the key information extracted from the image:

- Gregg hits the ball off the ground.
- The ball reaches a **maximum height** of 32 yards.
- The ball lands **170 yards away** horizontally.

### Question (a):
You are asked to model the path of Gregg's golf ball using a **quadratic equation**. Since projectile motion forms a parabola, we will write the equation in vertex form:
\[ y = a(x - h)^2 + k \]
where:
- \( h \) is the horizontal distance at which the maximum height occurs.
- \( k \) is the maximum height.
- \( a \) is a constant determining the concavity and steepness of the parabola.

From the problem:
- The ball reaches its maximum height of 32 yards at the midpoint of its horizontal range. Therefore, \( h = \frac{170}{2} = 85 \) yards.
- The maximum height \( k = 32 \) yards.
- The ball starts at the origin \((0,0)\) and ends at \((170,0)\).

### Finding \( a \):
At \( x = 0 \), the height \( y = 0 \), so we substitute into the equation to find \( a \):
\[
0 = a(0 - 85)^2 + 32
\]
\[
0 = a(85)^2 + 32
\]
\[
a = -\frac{32}{85^2}
\]

Now we can write the equation of the parabola:
\[
y = -\frac{32}{7225}(x - 85)^2 + 32
\]

This is the quadratic equation modeling the path of the ball.

Would you like more detailed steps or explanations?

Here are some follow-up questions:
1. Can you solve for the constant \( a \) in more detail?
2. How would you verify the equation by plugging in specific values for \( x \)?
3. What other forms can this quadratic equation take, such as standard form?
4. How would the graph of the parabola change if the maximum height were different?
5. What if the ball landed at a different horizontal distance?

**Tip:** Remember that in projectile motion problems, symmetry is key—the maximum height occurs halfway through the horizontal distance.

Gregg likes to golf and his favorite club is a 7 iron. While practicing at the range, Gregg hits the ball off the ground and it lands 170 yards away. Gregg’s ball reaches a maximum height of 32 yards. (a) Model the path of Gregg's golf ball with an appropriate quadratic equation. Write at least two forms.

Learn how to model the path of a golf ball hit by Gregg using a quadratic equation. The ball reaches a maximum height of 32 yards and lands 170 yards away. This problem involves understanding projectile motion and using quadratic equations in both vertex and standard form. Suitable for Grades 9-12.

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