To solve the given problem, let's start by determining the opening direction of the quadratic function $g(p) = 3p(4 - 5p)$.

Step 1: Determine the Opening Direction of the Quadratic

The quadratic function $g(p) = 3p(4 - 5p)$ can be written in standard form as:

$g(p) = 3p(4 - 5p)$

First, expand this expression to put it in the standard form of a quadratic, which is $g(p) = ap^2 + bp + c$.

Step 2: Expand the Expression Using the Area Model

To use the area model, we will distribute the terms:

$g(p) = 3p(4) - 3p(5p)$

Calculating each term:

$g(p) = 12p - 15p^2$

Now, let's rewrite it in standard form:

$g(p) = -15p^2 + 12p$

Step 3: Identify the Coefficients and Predict the Opening Direction

In the standard form $g(p) = ap^2 + bp + c$, the coefficient $a = -15$, $b = 12$, and $c = 0$.

Since the coefficient $a = -15$ is negative, the quadratic function opens downward.

Conclusion

• Opening Direction: Downward.
• Standard Form: $g(p) = -15p^2 + 12p$, which confirms the prediction that the quadratic opens downward due to the negative coefficient of $p^2$.

Would you like further details or have any questions?

Here are 5 related questions that expand upon this topic:

1. How does the coefficient of $p^2$ in a quadratic equation affect the shape of the graph?
2. What is the vertex form of a quadratic equation, and how can it be derived from the standard form?
3. How can we find the vertex of the quadratic function $g(p) = -15p^2 + 12p$?
4. What are the x-intercepts (roots) of the quadratic function $g(p) = -15p^2 + 12p$?
5. How does the quadratic formula help in solving quadratic equations?

Tip: The sign of the coefficient of $x^2$ in a quadratic equation determines whether the parabola opens upwards (positive) or downwards (negative).