Question

use a quadratic function to model the area of the rectangle. graph the function
2x + 6
x + 4
...
use the graphing tool to graph the function.
click to enlarge graph

Explanation:

Step1: Recall area of rectangle formula

Area = length $\times$ width

Step2: Substitute given side lengths

$A(x)=(2x+6)(x+4)$

Step3: Expand the product

$$\begin{align*} A(x)&=2x(x+4)+6(x+4)\\ &=2x^2+8x+6x+24\\ &=2x^2+14x+24 \end{align*}$$

Step4: Find key points for graphing

• Y-intercept: Set $x=0$, $A(0)=2(0)^2+14(0)+24=24$, so $(0,24)$
• X-intercepts: Set $A(x)=0$:

$2x^2+14x+24=0$
Divide by 2: $x^2+7x+12=0$
Factor: $(x+3)(x+4)=0$, so $x=-3, x=-4$

• Vertex: Use $x=-\frac{b}{2a}=-\frac{14}{2(2)}=-3.5$

Substitute $x=-3.5$: $A(-3.5)=2(-3.5)^2+14(-3.5)+24=-0.5$, so vertex at $(-3.5, -0.5)$

Step5: Plot and draw the parabola

Plot the intercepts and vertex, then draw a smooth upward-opening parabola through these points.

Answer:

The quadratic function is $A(x)=2x^2+14x+24$, and its graph is a parabola opening upwards with vertex at $(-3.5, -0.5)$, x-intercepts at $x=-3$ and $x=-4$, and y-intercept at $(0,24)$.