Predicting the Opening Direction:

We begin by predicting the opening direction of the quadratic function. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where the coefficient $a$ determines the opening direction.

• If $a > 0$, the parabola opens upward.
• If $a < 0$, the parabola opens downward.

The given quadratic is $f(x) = (x + 3)(2x - 1)$. Let's determine the sign of the leading term (which contains $x^2$) by multiplying the highest degree terms from each factor:

$x \times 2x = 2x^2$

Since the coefficient of $x^2$ is positive ($2$), we predict that the parabola opens upward.

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Using the Area Model to Express in Standard Form:

Now, let's expand $f(x) = (x + 3)(2x - 1)$ to get the quadratic in standard form $f(x) = ax^2 + bx + c$.

Using the distributive property (also known as the area model), we multiply each term in the first binomial by each term in the second binomial:

$f(x) = (x + 3)(2x - 1) = x(2x - 1) + 3(2x - 1)$

Now, perform the multiplication:

$x(2x - 1) = 2x^2 - x$ $3(2x - 1) = 6x - 3$

Now, combine all the terms:

$f(x) = 2x^2 - x + 6x - 3$

Simplify the like terms:

$f(x) = 2x^2 + 5x - 3$

Thus, the quadratic function in standard form is:

$f(x) = 2x^2 + 5x - 3$

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Verifying the Prediction:

Since the coefficient of $x^2$ is $2$, which is positive, the parabola indeed opens upward, as predicted.

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Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What does the constant term in the standard form represent geometrically?
2. How does the sign of the linear coefficient $b$ affect the symmetry of the parabola?
3. Can we predict the vertex location before expanding the quadratic?
4. How does factoring relate to the solutions of the quadratic equation?
5. What happens to the graph of the quadratic if the linear term is zero?

Tip: Always check the leading coefficient $a$ to quickly determine the direction of a parabola.