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Chapter 7 Graphs, Functions, and Linear Systems Slide 2 7.1 Graphing and Functions Slide 3 Objectives 1. Plot points in the rectangular coordinate system. 2. Graph equations in the rectangular coordinate system. 3. Use function notation. 4. Graph functions. 5. Use the vertical line test. 6. Obtain information about a function from its graph. Slide 4 Cartesian Coordinate System Rene Descartes (1596 1650) Invented Analytica Geometry Combined geometry and algebra Described shapes using algebraic expressions View relationships between numbers as graphs Describe shapes with equations. E.g., Line: y = 2x 1 Circle: x 2 + y 2 = 3 Parabola: y = 2x 2 + 3x - 1 Slide 5 Graphing Points A(2, 3) B(-2, 3) C(3, 2) D(-3, -2) Slide 6 Example Graph (-5, 3) and (3, -5) Slide 7 Example Plot the points: 1. A(3, 5) 2. B(2, 4) 3. C(5,0) 4. D(5,3) 5. E(0, 4) 6. F(0, 0). Slide 8 Graph of Equation Given: y = 4 x 2 Solution 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 x 2 } Graph of y = 4 x 2 Set of points which satisfy the equation. Slide 9 Graph of a Line Slide 10 Graph of a Line (cont.) y = x + 15 S = {(x, y) | y = x + 15} Slide 11 Functions Equation: 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 y Function: A rule for generating a value (for a dependent variable) from another value (independent variable) f(x) Slide 12 Functions If 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. Slide 13 Example Graph functions, for -2 x 2 f(x) = 2x g(x) = 2x + 4 Slide 14 Example (cont.) Slide 15 Vertical Line Test for Functions IF 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) Slide 16 Example: Analyzing a Graph The 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. a. What is the temperature at 8 a.m.? b. During which period of time is your temperature decreasing? c. Estimate your minimum temperature during the time period shown. How many hours after 8 a.m. does this occur? At what time does this occur? Slide 17 Example (cont.) During which period of time is the temperature increasing? Explain what is happening during 5 x 8. Explain why the graph defines y as a function of x.