Question

use a quadratic function to model the area of the rectangle. graph the function.
3x + 12
x + 2
...
the function is f(x) = 3x² + 18x + 24. (simplify your answer.)
use the graphing tool to graph the function.
click to enlarge graph

Explanation:

Step1: Factor out the common factor

The function is \( f(x) = 3x^2 + 18x + 24 \). First, factor out the greatest common factor, which is 3. So we get \( f(x) = 3(x^2 + 6x + 8) \).

Step2: Factor the quadratic inside the parentheses

Now, factor the quadratic \( x^2 + 6x + 8 \). We need two numbers that multiply to 8 and add up to 6. Those numbers are 2 and 4. So, \( x^2 + 6x + 8=(x + 2)(x + 4) \).

Step3: Write the simplified function

Substituting back, we have \( f(x)=3(x + 2)(x + 4) \). If we want to expand it (though factored form is simplified), we can also write it as \( f(x)=3x^2 + 18x + 24 \) (but factoring is a form of simplification here).

Answer:

The simplified form of the function \( f(x) = 3x^2 + 18x + 24 \) is \( f(x)=3(x + 2)(x + 4) \) (or expanded as \( f(x)=3x^2 + 18x + 24 \) if we consider the original form with the common factor factored out, but factored form is more simplified for analysis like graphing roots etc.). For graphing, the vertex form can also be found:

Step1: Complete the square

Start with \( f(x)=3x^2 + 18x + 24 \), factor out 3 from the first two terms: \( f(x)=3(x^2 + 6x) + 24 \).
To complete the square inside the parentheses, take half of 6 (which is 3), square it (which is 9), so add and subtract 9 inside the parentheses: \( f(x)=3(x^2 + 6x + 9 - 9) + 24 \).

Step2: Rewrite as a perfect square

This becomes \( f(x)=3((x + 3)^2 - 9) + 24 \). Distribute the 3: \( f(x)=3(x + 3)^2 - 27 + 24 \).

Step3: Simplify

Simplify the constants: \( f(x)=3(x + 3)^2 - 3 \).

So the vertex form (which is also a simplified form for graphing, showing vertex at \((-3, -3)\)) is \( f(x)=3(x + 3)^2 - 3 \).

The simplified function (in vertex form for graphing clarity) is \( \boldsymbol{f(x)=3(x + 3)^2 - 3} \) (or factored form \( \boldsymbol{f(x)=3(x + 2)(x + 4)} \)).