Chapter 3 Solving By Graph

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Chapter 3 Solving By Graph

  1. 1. 2.1 Functions and their Graphs p. 67
  2. 2. Assignment • Pp. 71-72 #5-48 all
  3. 3. Relations • A relation is a mapping, or pairing, of input values with output values. • The set of input values is called the domain. • The set of output values is called the range. • A relation as a function provided there is exactly one output for each input. • It is NOT a function if at least one input has more than one output
  4. 4. Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the Function? set notation!!! No: input 1 is mapped onto Both -2 & 1
  5. 5. Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output
  6. 6. A Relation can be represented by a set of ordered pairs of the form (x,y) Quadrant II X<0, y>0 Quadrant I X>0, y>0 Origin (0,0) Quadrant III X<0, y<0 Quadrant IV X>0, y<0
  7. 7. Graphing Relations • To graph the relation in the previous example: • Write as ordered pairs (-3,3), (1,-2), (1,1), (4,4) • Plot the points
  8. 8. (-3,3) (4,4) (1,1) (1,-2)
  9. 9. Same with the points (-3,3), (1,1), (3,1), (4,-2)
  10. 10. (-3,3) (1,1) (3,1) (4,-2)
  11. 11. Vertical Line Test • You can use the vertical line test to visually determine if a relation is a function. • Slide any vertical line (pencil) across the graph to see if any two points lie on the same vertical line. • If there are not two points on the same vertical line then the relation is a function. • If there are two points on the same vertical line then the relation is NOT a function
  12. 12. Use the vertical line test to visually check if the relation is a function. (-3,3) (4,4) (1,1) (1,-2) Function? No, Two points are on The same vertical line.
  13. 13. Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line
  14. 14. Graphing and Evaluating Functions • Many functions can be represented by an equation in 2 variables: y=2x-7 • An ordered pair is a solution if the equation is true when the values of x & y are substituted into the equation. • Ex: (2,-3) is a solution of y=2x-7 because: • -3 = 2(2) – 7 • -3 = 4 – 7 • -3 = -3
  15. 15. • In an equation, the input variable is called the independent variable. • The output variable is called the dependent variable and depends on the value of the input variable. • In y=2x-7 ….. X is the independent var. Y is the dependant var. • The graph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation.
  16. 16. Graphing an equation in 2 variables 1. Construct a table of values 2. Graph enough solutions to recognize a pattern 3. Connect the points with a line or curve
  17. 17. Graph: y = x + 1 Step 3: Step2: Step 1 Table of values
  18. 18. Function Notation • By naming the function ‘f’ you can write the function notation: • f(x) = mx + b • “the value of f at x” • “f of x” • f(x) is another name for y (grown up name) • You can use other letters for f, like g or h
  19. 19. Decide if the function is linear. Then evaluate for x = -2 • • • • f(x) = -x2 – 3x + 5 Not linear…. f(-2) = -(-2)2 – 3(-2) + 5 f(-2) = 7 • • • • • • g(x) = 2x + 6 Is linear because x is to the first power g(-2) = 2(-2) + 6 g(-2) = 2 The domain for both is….. All reals
  20. 20. Assignment

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