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 formula for rectangle

The area \( A \) of a rectangle is given by the product of its length and width. Here, the length is \( 3x + 6 \) and the width is \( x + 4 \). So, the function \( f(x) \) representing the area is the product of these two binomials: \( f(x)=(3x + 6)(x + 4) \).

Step2: Expand the product using distributive property (FOIL method)

First, multiply the First terms: \( 3x\times x = 3x^{2} \).
Then, the Outer terms: \( 3x\times4 = 12x \).
Next, the Inner terms: \( 6\times x = 6x \).
Finally, the Last terms: \( 6\times4 = 24 \).
Now, combine like terms (the \( x \)-terms): \( 12x+6x = 18x \).
So, putting it all together, \( f(x)=3x^{2}+12x + 6x+24=3x^{2}+18x + 24 \).

Answer:

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