Understanding applications of quadratic functions

Understanding applications of quadratic functions

by Karine Ptak

​Read the text of the example on the right

​The first thing we need to do is to write an equation.

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​What do we know? Area is equal to the product of the side measurements of the rectangle. How do we determine the measurements of the sides?

Fill in the Blank

The measurements of the sides of the larger rectangle are... 

Please separate the two expressions with the word "and" and do not use a space between the characters in your expressions.

​The correct answer contains 2x+11 (=length) and 2x+6 (=width).

​Please write this down, as you will need it later.

​Let's write the model

Do we agree that area of the outside/larger rectangle - area of the inside/smaller rectangle = area of the frame?

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​What is the area of the larger rectangle?

Fill in the Blank

The area of the entire rectangle is... Please write the model A=lw using the expressions.  Do not include spaces between characters.

A=...

​Please write it down since you will need it.

Fill in the Blank

The correct answer is similar to $$A=\left(2x+11\right)\left(2x+6\right)$$  . Please write the same model in standard form. Please use ^2 and no space between characters. You can watch the video to the left for help.

Please write it down since you will need it.​

Multiple Choice

What is the area of the inside/smaller rectangle? 

• 17

• 34

• 66

Fill in the Blank

Now find the difference between the areas of the two rectangles; simplify your answer, and keep it in standard form.

A=area of larger rectangle - area of smaller rectangle.

​Once again, write it down.

Fill in the Blank

What is the area of the frame supposed to be equal to?

​If we replace A by its value, we now have the equation

Fill in the Blank

Use one of the methods you have learned over the past week to solve this equation.  You may also use Desmos to graph the equation as a quadratic and linear system and find the intersection points.  Remember, however, that we are looking for x-values (the zeros/solutions).  When you are ready to move on, type "yes" in the blank below.

Multiple Choice

The solutions (zeros) to the equation $$4x^2+34x=28$$  are

• x=-2 and x=-11

• x=-9.26 and x=0.76

• (-2,0) and (-11,0)

• (-9.26,0) and (0.76,0)

​Since we are looking for which values of x will make the function = 0, the correct answers are the x-values only.

​x=-9.76 and x=-0.76

​Almost there!

​Watch the video attached to this slide to take a brain break.

​We know have to analyze which of the two solutions make sense for our story...

​Please substitute each value of x, one at a time, in both the length and the width measurements. Here they are again:

​

​f(x)=2x+11 and ​g(x)=2x+6

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​In other words, calculate f(-9.26), g(-9.26), f(0.76) and g(0.76). What do you notice???

​How do you know you're ready to move on?

​Open the assignment titled "I can create and use a quadratic model". Your submission is as follows:

​1. State how you chose to solve the equation in detail (you may submit a picture of your work)

​2. State which of the two zeros (if any) you would keep as a REASONABLE solution and why. Explain all of that in the context of the problem.