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. 1. f(x) = (x + 3)(2x – 1)

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.

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. 1. f(x) = (x + 3)(2x – 1)

Learn how to predict the opening direction of a quadratic function using the area model to express it in standard form. This example focuses on the quadratic function f(x) = (x + 3)(2x - 1) and explores polynomial expansion and graphing parabolas. Suitable for students in Grades 9-10.

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