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Linear and Quadratic Models

Straight lines and parabolas used to represent situations.

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Understand the concept of a function and use function notation. MCC9-12.FIF.1, 2, 3


MCC9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 

MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1 (n is greater than or equal to 1).

Learning to express real-life situations as mathematical functions allows seemingly complex ideas and actions to be broken down into smaller, simpler parts and analyzed.

How might you express the following mathematically?

Two brothers decide to race home from school, taking different routes. The second brother leaves 5 minutes after the first, and both arrive at home at the same time.

See if you can write an appropriate expression before the review at the end of the lesson that explains the process.

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Linear Models

The simplest functions are generally linear models. For instance, the equation y = 3x could be used to represent how much money you would bring in if you sold x boxes of cookies for $3 per box. Many situations can be modeled with linear functions. The key idea is that some quantity in the situation has a constant rate of change. In the cookie-selling example, every box costs $3.00. Therefore the profits increase at a constant rate. In sum, linear functions are used to model a situation of constant change, either increase or decrease.


You run a lawn mowing business, and charge $15 per lawn. Write a linear function to describe the amount of money made as a function of the number of lawns mowed.


If you express the number of lawns mowed as l, then $15 multiplied by l would represent the total money made based on the number of lawns, therefore:

15(l) = total income.


Solution to start-of-lesson question Did you come up with a mathematical model for the question at the start of the lesson?

Two brothers decide to race home from school, taking different routes. The second brother leaves 5 minutes after the first, and both arrive at home at the same time.

There are a number of different ways to model the information, depending on what part(s) of the information you choose to use. A couple of examples include:

If t = the time the second brother took to get home, then t + 5 = the time the first brother took.

If t = the time the first brother took to get home, then (t + (t - 5))/2 represents the average time to run home.

Your model may be similar, or may be written differently, but should compare different values given in the story problem.


A linear model is a function representing a situation involving a constant rate of change. The graph of a linear equation is a straight line.

Guided Practice


1) The Senior class has paid $1000.00 to a DJ for the Senior Prom. Tickets for the dance are $15.00 each.

a) Express the net income as a function of the number of tickets sold.
b) Graph the function and identify any limitations on the domain.

2) The Arlington Freshmen class wants to have a fundraiser. The class wants to buy a number of $4.00 flip-flops and $5.00 baseball hats, and has a total of $100 to spend.

a) If f represents the number of flip-flops and b represents the number of baseball hats, write a function to represent the number of flip-flops purchased as a function of unspent monies from baseball hats.
b) Using your equation from (a), determine the number of baseball hats that can be bought if 10 flip flops were purchased.

3) Studies of the metabolism of alcohol consistently show that blood alcohol content (BAC) declines linearly, after rising rapidly after initial ingestion. In one study, BAC in a fasting person rose to about 0.018 % after a single drink. After an hour the level had dropped to 0.010 %.

a) Write an equation relating BAC to time in hours after drinking t.
b) Assuming that BAC continues to decline linearly (meaning at a constant rate of change), approximately when will BAC drop to 0.002%?


1) To find the net income:

a) Let I(n) = net income from n tickets sold. Then the income is found by subtracting the cost of the DJ from the earned money. \begin{align*}I(n) = $15 \cdot n - $1000\end{align*}I(n)=$15n$1000
b) The domain of the function is limited to positive numbers, since the Senior Class will not sell a negative number of tickets. We can say, therefore, that the domain of the function is \begin{align*}n > 0\end{align*}n>0, where n is an integer.

2) To express this information as a function, remember that the question specified that there was $100 to spend, and that any money not spent on hats (at $5 ea) was spent on flip-flops (at $4 ea).

a) \begin{align*}\therefore f = \frac{$100 - 5h}{4}\end{align*}f=$1005h4
b) To calculate how many hats could be bought if 10 pairs of flip flops were purchased, substitute 10 in for f, and solve for h:
\begin{align*}10 = \frac{100 - 5h}{4}\end{align*}10=1005h4
\begin{align*}40 = 100 - 5h\end{align*}40=1005h
\begin{align*}5h = 60\end{align*}5h=60
\begin{align*}h = 12\end{align*}h=12

Therefore, if 10 pairs of flip flops were purchased, there would be money left over to buy 12 baseball caps.

3) In order to answer the question, you must express the relationship as an equation and then use the equation.

First, define the variables in the function and create a table.

The two variables are time and BAC.

Time BAC
0 0.018%
1 0.010%

Next, calculate the rate of change.

Time BAC Rate of change
0 0.018% 0
1 0.010% (0.008%)

This rate of change means when the time increases by 1, the BAC decreases (since the rate of change is negative) by .008. In other words, the BAC is decreasing .008% every hour. Since we are told that BAC declines linearly, we can assume that figure stays constant.

a) Now we can write an equation with b representing BAC and tthe time in hours:
\begin{align*}b = -.008t + .018\end{align*}b=.008t+.018
b) To learn when will the BAC reach .002%, substitute .002 in for b and solve for t.
\begin{align*}.002 = -.008t + .018\end{align*}.002=.008t+.018
\begin{align*}-.016 = -.008t\end{align*}.016=.008t
\begin{align*}t = 2\end{align*}t=2
Therefore the BAC will reach .002% after 2 hours.


  1. From 2002 - 2009 the number of gas stations in a certain country increased by 100 stations per year. In 2004 there were 1100 gas stations. Write a linear equation for the number of gas stations, (y), as a function of time, (t,) where t = 0 represents the year 2002.

Evaluate each function given the specified value.

  1. \begin{align*}Find f (-3) given f(x) = 2x^2 - 6x + 11\end{align*}Findf(3)givenf(x)=2x26x+11
  2. \begin{align*}Find f (2) given f (x) = 3(x+5)^2 - 2\end{align*}Findf(2)givenf(x)=3(x+5)22
  1. A rock is thrown from the top of a 763ft tall building. The distance, in feet, between the rock and the ground t seconds after it is thrown is given by \begin{align*}d = -16t^2 - 2t + 763\end{align*}d=16t22t+763 . How far from the ground is the rock after five seconds has elapsed?  eight seconds? nine seconds?  What can you conclude from your evaluations?
  2. Use the vertical motion formula \begin{align*}h = -16t^2 + v_0 t + s\end{align*}h=16t2+v0t+s to find the number of seconds it takes for a rocket launched from the ground with a starting velocity of 96 ft/s to reach an altitude of 45 ft. Round answers to the nearest tenth.  (Hint:  choose values for t and evaluate; record tested values-input and results-output in a table)
  3. The function \begin{align*}P = 0.0089t^2 +1.1149t + 78.4491\end{align*}P=0.0089t2+1.1149t+78.4491 models the United States population in millions since 1900. Use the function to predict the population in 1952.

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