For Questions 1–3, predict the opening direction of the quadratic and then use the area model to express the quadratic function in standard form f(x) = ax 2 + bx + c in order to verify your prediction. g(p) = 3p(4 – 5p) 

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).

For Questions 1–3, predict the opening direction of the quadratic and then use the area model to express the quadratic function in standard form f(x) = ax^2 + bx + c in order to verify your prediction. g(p) = 3p(4 – 5p)

Predict the opening direction of the quadratic function g(p) = 3p(4 – 5p) and express it in standard form f(x) = ax^2 + bx + c using the area model. This problem involves key concepts in algebra, including quadratic equations and function behavior, suitable for students in Grades 8-10.

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