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Comparing Linear, Quadratic, and Exponential Growth

Comparing Linear, Quadratic, and Exponential Growth

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Author: Jeremy Stephens
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Linear, Quadratic, and Exponential Models F-LE

Construct and compare linear, quadratic, and exponential models and solve problems. [Include quadratic.]

 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Linear Functions are of the form: f(n) = f(1) + d(n-1)

Linear functions have a common difference, d, which means you add the same amount to get from one term to the next term.

Quadratic Functions are of the form: f(n) = ax^2 + bx + f(0)

Quadratic functions have a common second difference, 2a, which means you increase the amount you add to each term by the same amount to get from one term to the next.

Exponential Functions are of the form: f(n) = f(1)*r^(n-1)

Exponential functions have a common ratio, r, which means you multiply by the same amount to get from one term to the next.

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Tutorial

Linear vs. Exponential Growth (Khan Academy)

Here is how you can distinguish between a linear and exponential function given a table of values. Linear relationships have a constant difference between terms in the output values given a constant difference in the input values. Exponential relationships have a constant ratio between the terms of the output values give a constant difference in the input values.

Linear relationship: f(x) = f(1) +d*(x-1)
In this equation the given variables represent the following: f(x) is the xth term in the pattern; f(1) is the 1st term in the pattern; d is the constant difference between consecutive terms in the pattern; x is which term we are looking at in the pattern(the 1st, 2nd, 3rd, 4th, etc.).

Exponential relationship: f(x) = f(1)*r^(x-1)
In this equation the given variables represent the following: f(x) is the xth term in the pattern; f(1) is the 1st term in the pattern; r is the constant ratio between consecutive terms in the pattern; x is which term we are looking at in the pattern(the 1st, 2nd, 3rd, 4th, etc.).

Source: khanacademy.org

Here is a real-life example that is not a perfect example of a linear relationship, but is super close to one.

Source: khanacademy.org

Linear vs Exponential Growth with MONEY

The difference between exponential and linear growth in terms of money.

Source: khanacademy.org

Linear vs Quadratic vs Exponential Growth

Now we add quadratic functions into the mix!
Quadratic functions have the form: f(x)=ax^2 + bx + c
We will explore how to find the equation of a quadratic function given a set of table values in another lesson. For now, you need to know that the second difference is constant between consecutive y terms in quadratic growth.

Source: CK-12 Foundation

x-axis = amount of time that has passed (in years)

y-axis = the number of professional contacts Jeremy has

Green = Emmanuel's approach

Blue = Quinn's approach

Red = Landon's approach


Let's say that Jeremy is just finishing college and is looking to expand his professional network of contacts. He currently has one professional contact. He gets advice from three different social gurus who know how make contacts in the professional world. Guru Landon tells him a way to make five professional contacts every year. Guru Quinn tells him a way to make 2 additional contacts within the first year, then 4 additional contacts the next year, then 6 additional contacts the next year, then 8 additional contacts the next year, and so on and so forth. Guru Emmanuel tells him a way to double the amount of contacts he has every year. 

Please pull up the Google Form and answer the given questions.

Source: desmos graphing calculator