Lesson 8-4 Modeling with Quadratic Functions

Lesson 8-4 Modeling with Quadratic Functions

By Laura Wroten

Bellringer

​What is the value of f(12)?

This simply means to plug in 12 everywhere you see x in the equation.

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Put it in your calculator now...​

Answer: A. 116

​Vocabulary

The equation h(t) = -16t²+v₀t+h₀ is the vertical motion model. The variable h represents the height of an object, in feet, t seconds after it is launched into the air. The term v₀ is the object's initial vertical velocity and h₀ is its initial height.

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Quadratic regression is a method used to find the quadratic function that best fits a data set. ​

Example 1 (Make sure you are looking at your notes)

Remember: A=LW

A. The length of the pool is 2x BUT there is also a 4ft wide deck on BOTH sides: L = 2x+8

The width of the pool is x BUT there is also a 4ft wide deck on BOTH sides: W = x+8

A=(2x+8)(x+8) -- Use the distributive property or box method to multiply.​

Multiple Choice

What is the simplified quadratic function?

f(x) = ___________________

• 2x²+24x+64

• 2x²+24x+16

• 2x²+64

• 2x²+16x+16

B. The paragraph tells us that the width of the pool is 15 feet, so we will plug 15 in for x.

​ f(15) = 2(15)²+24(15)+64

Put this in your calculator now to find the answer!​

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Multiple Select

Try It! 1. What will the new quadratic function be? Select all that apply.

• (3x+8)(x+8)

• (3x+4)(x+4)

• 3x(x+8)

• 3x²+32x+64

​Example 2: h(t) = -16t²+v₀t+h₀

A. What do we know from the problem?

v₀=16 (initial velocity)

h₀=30 (initial height - from picture)

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Plug those numbers into the equation to create your new equation.​

Multiple Choice

What is the new equation?

• h(t) = -16t²+30t+16

• h(t) = -16t²+16+30

• h(t) = -16t²+16t+30

• h(t) = -16t²+30+16

Multiple Choice

What is the x-value of the vertex?

• $$-\frac{1}{2}$$  

• $$\frac{1}{2}$$  

• 16

• -2

Once we know that the x-value of our vertex is 1/2, we can plug that in to the equation to find our y-value.

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Put this in your calculator now: -16(1/2)²+16(1/2)+30

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Multiple Choice

What is the y-value of the vertex?

• 1/2

• -34

• 34

• 75

The ordered pair of the vertex is at (1/2, 34).

*Remember, our x values are t (time) in this function. Our y values are h (height).​

This means that the diver reached the highest point of the dive, 34 feet (h), at 1/2 second (t).​

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He started at 30 feet, so subtracting 34-30 puts him at 4 feet above the platform.​

Try It!​

2. We will plug in 20 as his initial height and 8 as his initial velocity.

   h(t) = -16t²+8t+20

This time, use desmos.com to graph this function and find his maximum height.

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Go on, I'll wait...​

Multiple Choice

What was the vertex of the equation h(t) = -16t²+8t+20?

• (0, 20)

• (1.4, 0)

• (-0.9, 0)

• (0.25, 21)

Since the vertex is at (0.25, 21) and our y-value represents height, then the Diver's maximum height is 21 feet.

Example 3

Just some reminders:

• A residual is the result of the (actual y - predicted y​).

• The actual y-value comes from the data.

• The predicted y-value comes from the best fit function.

• A residual plot shows a good fit if the points are random and do not form a pattern. ​

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Subject | Subject

Some text here about the topic of discussion

Take a few minutes to finish graphing the actual data for Example 3. These points form a quadratic pattern. That's why the best fit function is a quadratic function.

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Now, find the residuals by completing the second table. Graph the residuals to create a residual plot.​

Example 4

Just like we can run a linear regression to find the line of best fit, we can run a quadratic regression when our data shows a quadratic pattern.

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Go to STAT: Edit: Put your x values (price increase) as L₁ and your y values (average revenue) as L_2.​

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Does your screen look like this?

​L	​L
​0	​745
​1	​846
​2	​910
​3	​952
​4	​1008

• Make sure you have Diagnostics on: Mode: STAT DIAGNOSTICS: ON

• STAT: CALC: 5 (QuadReg): Enter

• Go down to calculate and hit enter again​

  • a = the a value of your quadratic best fit function

  • b = the b value of your quadratic best fit function

  • c = the c value of your quadratic best fit function

  • r² tells you if the regression is a good fit. You want your r² value to be close to 1.​

Multiple Select

What is the quadratic function of best fit for this data and is it a good fit?

• ax²+bx+c

• -8+95.2+749.8

• -8x²+95.2x+749.8

• Yes; r² = 0.99 which is close to 1

• No; r² = 0.99 which is not close to 1

Now to answer the question on the paper...

If the price goes up one more time, our new x-value will be 5. Put your new function -8x²+95.2x+749.8 into y= and find y when x = 5.

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The predicted value of y = 1025.8, meaning the revenue would be $1025.80. That means the revenue will continue to go up if prices are increased one more time. ​

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Try it!

4. Stay in the table that you just used to answer the last question. Find y when x=6 and when x=7

What do you notice?

​x	​y
​6	​1033
​7	​1024.2

The revenue continued to increase with the 6th price increase, from $1025.80 to $1033. However, after the 7th price increase, the revenue started going down, to $1024.20.

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(What this means is that people will stop buying tickets once the price goes up so many times. Less tickets = less revenue)​

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Here's what we learned today...

• Area: When the length and width are represented by variable expressions, area will be represented by a quadratic function.

• Vertical Motion: The constant value in the vertical motion model gives us the initial height of the object.

• Data: We can use quadratic regression to find the best fit function for a set of quadratic data. A good fit means the r² value will be close to 1.​

What's next???

• Make sure you have all your notes filled out.

• In Google classroom, go to the section "Savvas Topic Practice". Click on the assignment labeled "Topic 8-4 Practice" and complete.