The problem involves modeling a flea's jump using the quadratic function:

$y = -0.073x(x - 33)$

Where:

• $x$ is the horizontal distance in centimeters, and
• $y$ is the corresponding height in centimeters.

You are asked to find two things:

1. How far did the flea jump?
2. What was the flea's maximum height?

Step 1: Finding the Total Jump Distance

The quadratic function is in factored form: $y = -0.073x(x - 33)$. To find how far the flea jumped, we solve for when $y = 0$ (this represents when the flea's height is 0, i.e., at the start and end of the jump).

Set $y = 0$: $-0.073x(x - 33) = 0$ This gives two solutions: $x = 0 \quad \text{(start of the jump)}$ and $x = 33 \quad \text{(end of the jump)}$

Thus, the flea jumped a total distance of 33 cm.

Step 2: Finding the Maximum Height

The maximum height of a parabola in this form occurs at the vertex. The x-coordinate of the vertex can be found by averaging the roots (0 and 33):

$x_{\text{vertex}} = \frac{0 + 33}{2} = 16.5$

Now, substitute $x = 16.5$ back into the original equation to find the maximum height:

$y = -0.073(16.5)(16.5 - 33)$ $y = -0.073(16.5)(-16.5)$ $y = -0.073 \times 16.5 \times -16.5 = 19.87 \, \text{cm}$

Thus, the maximum height of the flea's jump is approximately 19.87 cm.

Final Answers:

1. The flea jumped 33 cm.
2. The maximum height was 19.87 cm.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the flea's jump change if the coefficient of $x^2$ were more negative?
2. Can you graph the function and verify the vertex and roots?
3. How do different initial conditions (like varying coefficients) affect projectile motion?
4. What are some real-life applications of modeling jumps or motions using parabolas?
5. Could this model be used to predict jumps for other animals or insects?

Tip: The vertex form of a parabola provides a quick way to find the maximum or minimum value without needing to factor the equation.