Question

use a quadratic function to model the area of the rectangle. graph the function.
the function is f(x)=\square.
(simplify your answer.)

Explanation:

Step1: Recall area of rectangle formula

The area \( A \) of a rectangle is given by the product of its length and width. Here, the length is \( 3x + 12 \) and the width is \( x + 2 \), so the function \( f(x) \) (representing the area) is \( f(x)=(3x + 12)(x + 2) \).

Step2: Expand the product

Using the distributive property (FOIL method), we have:
\[

$$\begin{align*} f(x)&=3x(x + 2)+12(x + 2)\\ &=3x\cdot x+3x\cdot 2+12\cdot x+12\cdot 2\\ &=3x^{2}+6x + 12x+24 \end{align*}$$

\]

Step3: Combine like terms

Combine the \( x \)-terms: \( 6x+12x = 18x \). So,
\[
f(x)=3x^{2}+18x + 24
\]

Answer:

\( 3x^{2}+18x + 24 \)