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We are tasked with finding the equation of a quadratic curve for the hole that the gardener is digging. The general form of a quadratic equation is \( y = ax^2 + bx + c \).

### Given Information:
- The deepest point of the hole is 60 cm, which occurs at \( x = 40 \). This means the vertex of the parabola is \( (40, -60) \).
- The total width of the hole is 80 cm, so it extends 40 cm to the left and right of the vertex, implying that the x-intercepts are at \( x = 0 \) and \( x = 80 \).

### Steps to Solve:
1. **Vertex form of a parabola:** Since we know the vertex of the parabola, the equation can be written as:
   \[
   y = a(x - 40)^2 - 60
   \]
   where \( a \) is the unknown coefficient we need to find.

2. **Use the x-intercepts:** At \( x = 0 \) and \( x = 80 \), the depth \( y \) is 0. We can substitute one of these points into the equation to find \( a \).

Let's substitute \( x = 0 \) and \( y = 0 \) into the vertex form equation:
\[
0 = a(0 - 40)^2 - 60
\]
\[
0 = a(1600) - 60
\]
\[
a(1600) = 60
\]
\[
a = \frac{60}{1600} = \frac{3}{80}
\]

3. **Final Equation:**
Now that we have \( a = \frac{3}{80} \), we can write the equation of the hole as:
\[
y = \frac{3}{80}(x - 40)^2 - 60
\]

### Convert to Standard Form:
To match the required form \( y = ax^2 + bx + c \), expand the equation:
\[
y = \frac{3}{80}(x^2 - 80x + 1600) - 60
\]
\[
y = \frac{3}{80}x^2 - \frac{240}{80}x + \frac{4800}{80} - 60
\]
\[
y = \frac{3}{80}x^2 - 3x + 60 - 60
\]
\[
y = \frac{3}{80}x^2 - 3x
\]

Thus, the equation of the hole is:
\[
y = \frac{3}{80}x^2 - 3x
\]

---

Would you like more details or have any questions about this solution?

Here are five related questions to consider:
1. How would the equation change if the hole were deeper?
2. What happens to the parabola if the width of the hole increases?
3. How do we determine the vertex of a parabola from its equation?
4. Can you explain the relationship between the coefficient \( a \) and the shape of the parabola?
5. How does the quadratic equation relate to real-world problems like this one?

**Tip:** When solving quadratic problems with vertex information, always use the vertex form \( y = a(x - h)^2 + k \), as it simplifies calculations significantly!

A gardener is digging a hole to plant a tree. The hole can be modelled as a quadratic curve as shown in the diagram. The deepest point of the hole is 60 cm and the width of the hole is 80 cm. Find the equation of the hole, giving your equation in the form y = ax^2 + bx.

Learn how to derive the quadratic equation of a hole modeled by a parabola. Given the deepest point at (40, -60) and the hole width of 80 cm, find the equation of the hole in the form y = ax^2 + bx. This problem is ideal for students in Grades 9-11 and involves key algebraic concepts like vertex form and quadratic equations.

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