Linear functions - modeling

Section 1-4: Linear Functions - Modeling

Try the quiz at the bottom of the page! go to quiz

A function describes the way two quantities are related. y = 2x + 1 is a function as it describes how y relates to x. Every y value is found by doubling the x value and adding 1. Function notation replaces y with f(x) (pronounced "f of x") and is written f(x) = 2x + 1. This allows us to use different letters to represent different functions. A linear function is a function that has the form f(x) = mx + b. Some examples are given below:

f(x) = 5x - 7	f is a linear function of x
g(t) = .7t - 5	g is a linear function of t
h(s) = -5s + 2	h is a linear function of s
w(x) = 7	w is a linear function of x called a constant function

Sample problem

The junior class has paid $100 to a disc jockey for a dance. Tickets for the dance are $4 each.

1) Express the net income as a function of the number of tickets sold.
Solution: Let I(n) = net income from n tickets sold. Then the income is found by subtracting the cost of the dj from the money earned.
I(n) = 4n - 100
2) Graph the function
Solution: n can only be a non-negative integer. The domain of the function is n > 0, where n is an integer. The graph looks as follows:

Notice that they lose $100 if they sell no tickets. The break-even point is selling 25 tickets. Why? They make money selling more than 25 tickets. This graph represents a mathematical model which is describing a real world situation. Notice that the graph is made up of individual points. Graphs of this nature are called discrete functions. If we connect the dots and form a line it is a continuous function. This is sometimes given as the graph, but keep in mind the values must be integers!

Suppose the phone company charges 20 cents the first minute and 10 cents for each additional minute rounded up to the next minute. (1.2 minutes would round to 2 full minutes). Graph a mathematical model of the cost of a call lasting t minutes.

Time in minutes t	Cost in cents C(t)
0 < t < 1	20
1 < t < 2	30
2 < t < 3	40
3 < t < 4	50

Look at the above graph. Does it model what is happening with the cost of a phone call? It seems to. Notice the open dots at the left end of each line segment. This indicates that at time t = 0, there is no charge because you haven't made a phone call. According to the graph, what is the charge at t = 1? It's 20 cents like it is supposed to be! A linear equation that would somewhat model this would be the equation C(t) = 10x + 20. The graph above is called a step function because it looks like a series of small steps! The linear graph looks as follows:

If you use this graph as the model, be aware that the estimates are over the actual cost because from point to point the change is constant(slope)!

Well pardner, it's time to mosey on to the next section!! On to complex numbers!! Imagine that! (Do you get the pun?)

Current quizaroo # 1a

a) (-1, -2); b) (2, 7); c) (1, 2); d) (4, 14); e) (-2, -4)
: b) Lines are perpendicular; c) Lines intersect but not at right angles; d) Lines coincide; e) Not enough info to tell
3) Write the equation of the line with slope = -2 and y-intercept = 5: a) 5x + y = 2; b) 5x + y = -2; c) 2x + y = -5; d) -2x + y = 5; e) 2x + y = 5

4) Find the equation of the line perpendicular to the line y = (1/2)x + 4 going through the point
(0,2): a) y = 2x; b) y = -2x; c) y = 2x - 2; d) y = -2x + 2; e) 2y = x + 4

5) Jack's new truck costs $325 per month for car payments. He estimates that gas and maintenance expenses cost $.23 per mile. Express the monthly cost as a function of the miles driven.

a) C(m) = 325m + .23

b) C(m) = -.23m + 325

c) C(m) = .23m + 325

d) C(m) = .23m - 325

e) C(m) = 325m - .23

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