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Chapter 7 Graphs, Functions, and Linear Systems

Chapter 7 Graphs, Functions, and Linear Systems. 7.1 Graphing and Functions. Objectives. Plot points in the rectangular coordinate system. Graph equations in the rectangular coordinate system. Use function notation. Graph functions. Use the vertical line test.

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Chapter 7 Graphs, Functions, and Linear Systems

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  1. Chapter 7Graphs, Functions, and Linear Systems

  2. 7.1 Graphing and Functions

  3. Objectives • Plot 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.

  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: x2 + y2 = 3 • Parabola: y = 2x2 + 3x - 1

  5. Graphing Points • A(2, 3) • B(-2, 3) • C(3, 2) • D(-3, -2)

  6. Example • Graph (-5, 3) and (3, -5)

  7. Example • Plot the points: • A(−3, 5) • B(2, −4) • C(5,0) • D(−5,−3) • E(0, 4) • F(0, 0).

  8. Graph of Equation • Given: y = 4 – x2 • 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 – x2} • Graph of y = 4 – x2Set of points which satisfy the equation.

  9. Graph of a Line

  10. Graph of a Line (cont.) y = x + 15S = {(x, y) | y = x + 15}

  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)

  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.

  13. Example Graph functions, for -2 ≤ x ≥ 2 f(x) = 2x g(x) = 2x + 4

  14. Example (cont.)

  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)

  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. • 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?

  17. 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.

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