Linear Functions and Models Complete the table of

Linear Functions and Models Complete the table of values below for: L = 0. 0001 T + 10 T (in C) | 0 | 10 | 20 | 30 | 40 | L (in m) | 10 | ? | ? | a) Graph the values found above using T as the horizontal axis and L as the vertical axis. b) The graph in part (a) looks very much like a ? ? (horizontal line – dah!) c) Why ? ? (slope is so small – very close in value to ? ? )

A function: -- describes a dependent relationship between quantities. For example the expression 3 x – 5 depends on the value of ‘x’, so we say that ‘ 3 x – 5’ is a function of ‘x’. In function notation we write: f(x) = 3 x – 5 (normal notation reads as: y = 3 x - 5 f(x) is read ‘f’ of ‘x’ Normal notation: states that y = 3 x – 5, so this allows us to say that ‘y = f(x)’ In function notation, if f(x) = 3 x – 5, then f(1) = ? ? 3(1) – 5 or f(1) = - 2 So when x = 1, f(x) (which is another name for ‘y’) = - 2 so we basically come up with (1, -2)

If f(a) = 0, then whenever x = a, this makes the function = 0. which gives the point (a, 0); therefore ‘a’ is called a zero of the function. Other examples of linear functions and their meaning: L(t) = 0. 0001 T + 10 g(s) = -1. 2 s + 4. 7 h(t) = 3 ‘L’ is a linear function of ‘T’ ‘g’ is a linear function of ‘s’ ‘h’ is a linear function of ‘t’ Examples: What is the slope of the graph of f(x) = -2 x + 4 ? ? ? - 2 a) What is a ‘zero’ of the function? Set -2 x + 4 = 0 and solve for x b) What are the intercepts of the functions graph? y-intercept is 4 and x-intercept is 2 same value as the zero! c) Find f(3/2) Replace (3/2) in for ‘x’ and simplify to find f(x) or ‘y’

Example: Let ‘g’ be a linear function such that g(1) = 2, and g(5) = 4 a) Sketch the graph of ‘g’ plot the ordered pairs (1, 2) & (5, 4) b) Find an equation for ‘g(x)’ Find the slope then use point-slope form) c) Find g(-1) Replace -1 in for ‘x’ in to your newly found equation Example: The senior class has paid $200 to rent a roller skating rink for a fund raising party. Tickets are $5 each. a) Express the ‘net income’ as a function of the number of tickets sold Let ‘x’ = Number of tickets sold. $200 –> is it + or - • Graph the function and identify a) the point at which the class b) begins to make a profit. c) This would be the ‘zero’ or ‘x’-intercept!

Example: Suppose that is costs 50 cents for the first minute of a long distance telephone call and 20 cents for each additional minute or fraction thereof. Give a graphical model of the cost of a call lasting ‘t’ minutes. 1 st: Lets analyze the information: time ‘t’ If 0 < t < 1 If 1 < t < 2 If 2 < t < 3 If 3 < t < 4 cost ? ? ? ? This type of problem actually models what is called a ‘step function’. Can you figure out why?

We could arrive at a ‘linear model’ to represent this situation. Would you say that ‘cost’ is a function of ‘time’, Or ‘time’ is a function of ‘cost’? ? ? C(t) = 50 + 20 t The problem with the algebraic model is that at times It may ‘over’ or ‘under’ estimate the cost. Talk about ‘domain’: and ‘range’: the set of all ‘input’ values or ‘x’s the set of all ‘outcomes’ or ‘y’s.