A jewelry maker has total revenue for her necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces produces and sold. How many necklaces must be produced and sold in order to break even?

A graph is a diagram of a relationship of (at least) two variables with changing values.

The current value of each variable is represented as a distance from the origin.The coordination of the variables is represented as a point on the graph

All points on the graph are equally important.

A graph can just be a set of discrete points

{(0,1), (1.5,3), (2.3,), (5,0.7), (5+,6), (9,2)}

Or a shape

The equation of a graphThe algebraic relationship between variables on a graphThe equation is test: If you pick a pair of values that makes the equation true, that point is on the graph. If you pick a pair that is not true, the point is not on the graph.Graphing is finding all the points that make the equation true, and changing their color.How would we find the equation of this graph?

Tool: The distance between two points

Tool: The distance between two points

Horizontal distance between points is x-a

Vertical distance between points is y-b

We have a right triangle.

Pythagorean Theorem

Distance Formula

The distance between (a,b) and (x,y) isBack to a circle problem

CircleA circle is the set of all points a given distance (the radius) from a given point (the center).Our circle is all the points 4 units away from (1,2)

All points 4 units away from (1,2)For any point (x,y)The distance between (1,2) and (x,y) is 4Using the Pythagorean Theorem

Is the equation of the circle

Equation of a circleFor radius r and center (a,b)

Consider a circle with equation x2 + (y+4)2 = 289. The center and radius are given by:(0,4), radius=289(0,-4), radius=289(4,0), radius=289(-4,0), radius=289None of the above. Consider a circle with equation x2 + (y+4)2 = 289. The center and radius are given by:

EMidpoint formulaThe point halfway between (a,b) and (x,y) is

FunctionsA function is a relationship between two changing variablesAn input variableAn output variableThe result of doing the function to the output variableBoth variables change so that the input variable always tells you exactly what the output variable is.You never get two outputs for the same input.

inputOutputFunctionAny time I know the input, thats enough information to tell me the output.

Example: when input is 2, output is 20.

OutputInputNot a FunctionJust knowing the input is not enough to tell me the output

Example: when input is 20, the output could be 2, 3.6, or 6.9.Function

inputoutput

Not a FunctioninputoutputFunction notationFor a function named And an input variable named xThe output variable is named (x).(x) is the number that is the result of doing the action to the number xWARNING(x) DOES NOT MEAN *xYou can only multiply numbers. f is NOT a number. f is the name of a relationship.x and (x) are the numbers.Brangelina is not a person.Brangelina is the name of a relationshipBrad and Angelina are the peopleHow to do a function to the input numberAlgebra: Substitute

How to do a function to the input number

x(x)Find your input value on the x axis. Here the input value is 5.5How to do a function to the input number

x(x)Go up and over to Find your output value on the (x) axis. Here the output value is 25How to do a function to the input number

x(x)Input 5.5, output 25, name of function . (5.5)=25.Given the function f(x)=x2 -2Mx, where M is some parameter, find f(3).3 3M9-6M 6-9MNone of the above. Given the function f(x)=x2 -2Mx, where M is some parameter, find f(3).

CDomain and RangeFunction is a relationship between a changing input variable and a changing output variable.The domain is a description of all the values that the changing input variable takes on.The range is a description of all the value that the changing output variable takes on.

x(x)ExampleDomain: [0,9]Range: [0,30]A)

B)

C)

D)

E) None of the above

State the range of the function whose graph is pictured here. Select the best answer!

DState the range of the function whose graph is pictured here. Select the best answer!

First, the output changes from -1 to -3 (excluding -1).The output can be any value between -3 and -1, but cant be -1. Range: [-3,-1)

Later, the output changes from 2 to 3.The output can be any value between 2 and 3, including 2 and 3. Additional Range: [2,3]