To predict the opening direction of the quadratic function $f(x) = (x + 3)(2x - 1)$, we first need to expand the expression and determine the value of $a$, the coefficient of $x^2$. The sign of $a$ will tell us the opening direction of the parabola:

• If $a > 0$, the parabola opens upwards.
• If $a < 0$, the parabola opens downwards.

Step 1: Expanding the quadratic using the area model

The area model is a visual representation of multiplying two binomials. In this case, we'll expand $(x + 3)(2x - 1)$.

We multiply each term of the first binomial by each term of the second binomial:

$(x + 3)(2x - 1) = x \cdot 2x + x \cdot (-1) + 3 \cdot 2x + 3 \cdot (-1)$

This gives:

$= 2x^2 - x + 6x - 3$

Step 2: Combine like terms

Now, simplify the expression by combining like terms:

$= 2x^2 + (6x - x) - 3$ $= 2x^2 + 5x - 3$

Step 3: Verify the prediction

Now that we have expressed the quadratic in standard form $f(x) = 2x^2 + 5x - 3$, we can identify the coefficient $a = 2$. Since $a > 0$, the parabola opens upwards.

Final Answer:

• The quadratic opens upwards because the coefficient $a = 2$ is positive.
• The quadratic in standard form is $f(x) = 2x^2 + 5x - 3$.

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Do you want more details or have any questions?

Here are 5 related questions to extend your understanding:

1. What is the vertex form of the quadratic function $f(x) = 2x^2 + 5x - 3$?
2. How do you find the axis of symmetry of a quadratic function?
3. What is the significance of the constant term $c$ in a quadratic function?
4. How would the graph change if $a$ was negative?
5. How can we determine the x-intercepts of a quadratic function?

Tip: The vertex of a parabola can be found using the formula $x = -\frac{b}{2a}$, which gives the x-coordinate of the vertex.