Functions and Systems of Equations Function Notation Systems of Linear Equations Solving using the Substitution Method Solving using the Elimination Method

Total Revenue for GE Find an equation for a model of these data. b) Using your model, estimate GE’s revenue in 2010 c)What is the slope of your model? What does it mean in regard to GE’s revenue

Relation = set of ordered pairs. Relation between elements of a set of input and elements of a set of outputs (4,11) x + 2y = 12

Function = every input there is only one output For each input value in the domain, you must have one and only one output value in the range. {(1,2) , (3,4) , (5, 6) , (7,8)} 1 4 5 7 9 3 4 5 6 9

Function or Not? The Set A = {(2,5), (4,8), ( 10,8), (20,15)} 90 88 91 Day of Week Monday Wednesday Saturday Temperature (degrees Fahrenheit) 90 88 91 93

Vertical Line Test Given the graph of an equation utilize a vertical line. Try to determine how many times the vertical line touches the equation. Vertical Line Test = any vertical line intersects a graph in at most one point, the graph represents a function

Function or Not? 𝑦=7𝑥 −20 𝑦 2 = 4𝑥 2 +16

Notation f(x) “f of x” Represents a function named f that depends on the variable x f means output/y-variable/range Shorthand method of providing information in a compact form.

Example 𝑃(𝑡) = population of Hawaii (in milliions) 𝑡 = Years since 2000 Then the population of Hawaii at time t can be represented by the following function. 𝑃 𝑡 =0.013𝑡+1.21

Word problems and Function Notation 𝐻(𝑡) = height of a toy rocket in feet t seconds after launch. What does 𝐻(3) = 12 mean? 𝐶(𝑚) = Cost in hundred of dollars for producing m miracle mops. 𝐶 (2500) = 189 𝑃(𝑡) = population of Michigan, in millions, t, years since 2000. 𝑃(10) = 10.4

Writing models in Function notation Define the variable Create Scatterplot Choose model type{linear, quadratic, poly….) Linear: Determine the slope and y-int Check model by graphing.

Population of Wisconsin, in millions Find an equation for a model of these data. Write your model in function notation. Determine a reasonable domain and range Find P(14) and interpret its meaning in regard to the pop of Wisconsin.

Using Function Notation 𝑓 𝑥 =7𝑥+2 𝑔 𝑥 =−1.25𝑥+14 ℎ 𝑥 =2 𝑥 2 −10 Find: 𝑓 3 ℎ 5 𝑔 𝑥 =15

Domain and Range In an application need to consider model breakdown. In an equation Domain: set of all real numbers that make the function defined. (Division by zero and imaginary numbers need to be avoided) Range: set of all possible output values resulting from values of the domain.

Systems of Linear Equations Set of two or more linear equation Solution: An ordered pair (or set) that is a solution to every equation in the system.