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Chapter 7Graphs, Functions, and Linear Systems7.1 Graphing and FunctionsObjectivesPlot points in the rectangular coordinate system.Graph equations in the rectangular coordinate system.Use function notation.Graph functions.Use the vertical line test.Obtain information about a function from its graph.Cartesian Coordinate SystemRene Descartes (15961650)Invented Analytica GeometryCombined geometry and algebraDescribed shapes using algebraic expressionsView relationships between numbers as graphs

Describe shapes with equations. E.g.,Line: y = 2x 1Circle: x2 + y2 = 3Parabola: y = 2x2 + 3x - 1

Graphing PointsA(2, 3)B(-2, 3)C(3, 2)D(-3, -2)

ExampleGraph (-5, 3) and (3, -5)

ExamplePlot the points: A(3, 5)B(2, 4)C(5,0)D(5,3)E(0, 4)F(0, 0).

Graph of EquationGiven: y = 4 x2Solution set of the equation: Set of all ordered pairs (x, y) which will make the equation true.Solution = {(x, y} | (x, y) satisfy the equation y = 4 x2}Graph of y = 4 x2Set of points which satisfy the equation.

Graph of a Line

Graph of a Line (cont.)

y = x + 15

S = {(x, y) | y = x + 15}FunctionsEquation: y = x + 15.We can say that the rule for obtaining y, given x, is: f(x) = x + 15.The notation y = f(x) indicates that the variable y is a function of x. The notation f(x) is read f of x.x yFunction: A rule for generating a value (for a dependent variable) from another value (independent variable)f(x)FunctionsIf an equation in two variables (x and y) yields precisely on value of y for each value x, then y is a function of x.y = f(x)y is a function of x.ExampleGraph functions, for -2 x 2 f(x) = 2x g(x) = 2x + 4

Example (cont.)

Vertical Line Test for FunctionsIF a vertical line intersects a graph in more than one point, the graph does not describe a function of x.Which of the following is a function?

a) b) c) d)

Example: Analyzing a GraphThe given graph illustrates the body temperature from 8 a.m. through 3 p.m. Let x be the number of hours after 8 a.m. and y be the body temperature at time x. What is the temperature at 8 a.m.?During which period of time is your temperature decreasing?Estimate your minimum temperature during the timeperiod shown. How many hours after 8 a.m. does this occur?At what time does this occur?

Example (cont.)During which period of time isthe temperature increasing?Explain what is happening during 5 x 8.Explain why the graph defines y as a function of x.