Question

use a quadratic function to model the area of the rectangle. graph the function
2x + 6
x + 1
...
use the graphing tool to graph the function
click to enlarge graph

Explanation:

Step1: Recall rectangle area formula

Area = length $\times$ width

Step2: Substitute given side lengths

$A(x)=(2x+6)(x+1)$

Step3: Expand the product

$$\begin{align*} A(x)&=2x(x+1)+6(x+1)\\ &=2x^2+2x+6x+6\\ &=2x^2+8x+6 \end{align*}$$

Step4: Find key graph points

• Vertex: Use $x=-\frac{b}{2a}=-\frac{8}{2\times2}=-2$. Substitute $x=-2$: $A(-2)=2(-2)^2+8(-2)+6=-2$. Vertex: $(-2,-2)$.
• X-intercepts: Set $2x^2+8x+6=0$, factor to $2(x+3)(x+1)=0$, so $x=-3, x=-1$.
• Y-intercept: Set $x=0$, $A(0)=6$.

Step5: Graph the parabola

Plot vertex, intercepts, and draw upward-opening parabola (since $a=2>0$). For context of area, we only consider $x>-1$ (since side lengths must be positive: $x+1>0 \implies x>-1$).

Answer:

The quadratic function is $A(x)=2x^2+8x+6$, and its graph is a parabola opening upwards with vertex at $(-2, -2)$, x-intercepts at $x=-3$ and $x=-1$, and y-intercept at $(0,6)$.