Modelling with Quadratics and Exponentials

Modelling with Quadratics and Exponentials

Modelling: What does it mean?

• Represent a real world situation via a literal model

• Represent said situation with equations (Yeah that happens)

• Different equations for different situations

Multiple Choice

Which statement below accurately represents the graph?

• $100 is earned for each hour worked
• $100 is earned for every 2 hours worked
• $500 is earned for every 8 hours worked
• $400 is earned for every 6 hours worked

Multiple Choice

The equation below represents an elevator's height after an amount of seconds. Using this equation, what should the height of the elevator be after 14 seconds?
h(s) = -3.8s + 290

• 80
• 236.8
• -80
• 72.6

Multiple Choice

Which of the following situations could be descried by the equation y=120-25x?

• There are 120 people in the football stadium, and 25 more are entering each hour.

• A plumber charges $25 for a house call and $120 per hour.

• A teacher has 120 students. Her third period has 25 students

• There are 120 gallons of water in a tank. It releases water at a rate of 25 gallons per minute

Linear Models

These have a very straight forward usage - the situation has a constant rate of change that affects the entire equation. We most commonly use slope-intercept form.

Quadratic Models

• Where do we use them?

• What purpose do they serve?

The function below describes the height in meters of a basketball x seconds after it has been thrown vertically into the air. 

 $f\left(x\right)=-4.9x^2+17x+0.6$  

Multiple Choice

When will the basketball reach it's maximum height?

• vertex (y-value)

• zero/x - intercept

• vertex (x-value)

Multiple Choice

What is the maximum height of the basketball?

• vertex (y-value)

• zero/x -intercept

• vertex (x-value)

Multiple Choice

When will the basketball hit the floor?

• vertex (y-value)

• zero/x -intercept

• vertex (x-value)

What are the zeros?

The function below describes the height in meters of a basketball x seconds after it has been thrown vertically into the air. 

 $f\left(x\right)=-4.9x^2+17x+0.6$  

Open Ended

What are the zeros?

 $0=-4.9x^2+17x+0.6$  

A bicycle company's profit can be represented by the equation below where P(x) is the profit in thousands of dollars and x is the time in years.

 $f\left(x\right)=-2x^2+92x-84$  

Open Ended

How many years before the company has a profit of 0?

 $f\left(x\right)=-2x^2+92x-84$  

Poll

Exponential functions use the terms "growth" and "decay" instead. Which of the two graphs do you think is "growing" from left to right, and could be labeled "growth"? Click on your choice.

Multiple Choice

Which of the graphs is exponential decay?

Multiple Choice

Use desmos.com/calculator to determine which table matches the function 

 $h\left(x\right)=12\left(\frac{1}{3}\right)^x$  

Multiple Choice

Use desmos.com/calculator to determine which table matches the function 

 $f\left(x\right)=2\left(3\right)^x$  

Multiple Choice

Is

 $f\left(x\right)=2\left(3\right)^x$  growth or decay? How do you know?

• Growth because it's going downwards from left to right.

• Growth because it's going upwards from left to right.

• Growth because the table of values is decreasing, showing a pattern of division.

• Growth because all exponential functions grow.

Open Ended

What do you notice about the examples of exponential growth functions? Is there a way you can determine an exponential function is "growing" instead of "decaying"?

Here are more examples of exponential functions that model "growth":

•  $f\left(x\right)=4^x$  

•  $g\left(x\right)=\frac{1}{2}\left(2\right)^x$  

•  $y=12\left(50\right)^x$  

•  $f\left(x\right)=2\left(1.5\right)^x$  

Multiple Choice

Is

 $h\left(x\right)=12\left(\frac{1}{3}\right)^x$  growth or decay? How do you know?

• Decay because it's going downwards from left to right.

• Decay because it's going upwards from left to right.

• Decay because the table of values is increasing, showing a pattern of multiplication.

• Decay because all exponential functions decrease.

Here are more examples of exponential functions that model "decay":

•  $f\left(x\right)=\left(\frac{1}{4}\right)^x$  

•  $g\left(x\right)=2\left(\frac{2}{3}\right)^x$  

•  $y=-12\left(0.9\right)^x$  

•  $y=10\left(\frac{1}{2}\right)^x$  

Open Ended

Closure: Respond to each question with a complete sentence. What do you notice is the same about "growth" functions and "decay" functions? What do you notice is different about "growth" functions and "decay" functions? How can you tell a function is "decaying" or "growing", without seeing the graph?