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FUNCTIONS AND GRAPHS

FUNCTIONS AND GRAPHS. Aim #1.2 : What are the basics of functions and their graphs?. Let’s Review: What is the Cartesian Plane or Rectangular Coordinate Plans? How do we find the x and y-intercepts of any function? How do we interpret the viewing rectangle [-10,10, 1] by [-10, 10,1]?.

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FUNCTIONS AND GRAPHS

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  1. FUNCTIONS AND GRAPHS

  2. Aim #1.2: What are the basics of functions and their graphs? • Let’s Review: • What is the Cartesian Plane or Rectangular Coordinate Plans? • How do we find the x and y-intercepts of any function? • How do we interpret the viewing rectangle [-10,10, 1] by [-10, 10,1]?

  3. IN THIS SECTION WE WILL LEARN: • How to find the domain and range? • Determine whether a relation is a function • Determine whether an equation represents a function • Evaluate a function • Graph functions by plotting points • Use the vertical line to identify functions

  4. WHAT IS A RELATION? • A relation is a set of ordered pairs. • Example: (4,-2), (1, 2), (0, 1), (-2, 2) • Domain is the first number in the ordered pair. Example:(4,-2) • Range is the second number in the ordered pair. • Example: : (4,30)

  5. EXAMPLE 1: • Find the domain and range of the relation. • {(Smith, 1.0006%), (Johnson, 0.810%), (Williams, 0.699%), (Brown, 0.621%)}

  6. PRACTICE: • Find the domain and range of the following relation.

  7. HOW DO WE DETERMINE IF A RELATION IS A FUNCTION? • A relation is a function if each domain only has ONE range value. • There are two ways to visually demonstrate if a relation is a function. • Mapping • Vertical Line Test

  8. DETERMINE WHETHER THE RELATION IS A FUNCTION • {(1, 6), (2, 6), (3, 8), (4, 9)} • {(6, 1), (6, 2), (8, 3), (9, 4)}

  9. FUNCTIONS AS EQUATIONS • Functions are usually given in terms of equations instead of ordered pairs. • Example: y =0.13x2 -0.21x + 8.7 • The variable x is known the independent variable and y is the dependent variable.

  10. HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION? • x2 + y = 4 • Steps: • Solve the equation for y in terms of x. • Note: • If two or more y values are found then the equation is not a function.

  11. HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION • x2 + y2 =4 • Steps: • Solve the equation for y in terms of x. • Note: • If two or more y values are found then the equation is not a function.

  12. PRACTICE: • Solve each equation for y and then determine whether the equation defines y as a function of x. • 2x + y = 6 • x2 + y2 = 1

  13. WHAT IS FUNCTION NOTATION? • We use the special notation f(x) which reads as f of x and represents the function at the number x. • Example: f (x) = 0.13x2 -0.21x +8.7 • If we are interested in finding f (30), we substitute in 30 for x to find the function at 30. • f (30)= 0.13(30)2 -0.21 (30) + 8.7 • Now let’s try to evaluate using our calculators.

  14. HOW DO WE EVALUATE A FUNCTION? • F (x) = x2 + 3x + 5 • Evaluate each of the following: • f (2) • f (x + 3) • f (-x) • Substitute the 2 for x and evaluate. • Then repeat.

  15. GRAPHS OF FUNCTIONS • The graph of a function is the graph of the ordered pairs. • Let’s graph: • f (x) = 2x • g (x) = 2x + 4

  16. USING THE VERTICAL LINE TEST • The Vertical Line Test for Functions • If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

  17. Practice: Use the vertical line test to identify graphs in which y is a function of x.

  18. SUMMARY: ANSWER IN COMPLETE SENTENCES. • What is a relation? • What is a function? • How can you determine if a relation is a function? • How can you determine if an equation in x and y defines y as a function of x? Give an example.

  19. AIM #1.2B: WHAT KIND OF INFORMATION CAN WE OBTAIN FROM GRAPHS OF FUNCTIONS? • Note the closed dot indicates the graph does not extend from this point and its part of the graph. • Open dot indicates that the point does not extend and the point is not part of the graph.

  20. HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A FUNCTION’S GRAPH? • Domain: set of inputs Found on x –axis Range: set of outputs Found on y -axis

  21. Using set builder notation it would look like this for the domain: • Using Interval Notation: [-4, 2] • What would it look like for the range using both set builder and interval notation?

  22. Use Set Builder Notation. Domain: Range: IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH

  23. IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH

  24. IDENTIFYING INTERCEPTS FROM A FUNCTION’S GRAPH • We can say that -2, 3, and 5 are the zeros of the function. The zeros of the function are the x- values that make f (x) = 0. Therefore, the real zeros are the x-intercepts. A function can have more than one x-intercept, but at most one y-intercept.

  25. SUMMARY: ANSWER IN COMPLETE SENTENCES. • Explain how the vertical line test is used to determine whether a graph is a function. • Explain how to determine the domain and range of a function from its graph. • Does it make sense? Explain your reasoning. I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.

  26. AIM #1.3: HOW DO WE IDENTIFY INTERVALS ON WHICH A FUNCTION IS INCREASING OR DECREASING? • Increasing, Decreasing and Constant Functions • A function is increasing on a open interval, I if f (x1) < f(x2) whenever x1<x2 for any x1 and x2 in the interval.

  27. A function is decreasing on an open interval, I, if f(x1) > f (x2) whenever x1 > x2 for any x1 and x2 in the interval.

  28. A function is constant on an open interval, I, f(x1) = f (x2) for any x1 and x2 in the interval.

  29. Note: • The open intervals describing where function increase, decrease or are constant use x-coordinates and not y-coordinates.

  30. Example 1: Increases, Decreases or Constant State the interval where the function is increasing, decreasing or constant.

  31. Practice: State the interval where the function is increasing, decreasing or constant.

  32. WHAT IS A RELATIVE MAXIMA? • Definition of a Relative Maximum A function value f (a) is a relative maximum of f if there exists an open interval containing a such that f (a) > f (x) for all x ≠ a in the open interval.

  33. WHAT IS A RELATIVE MINIMA? • Definition of a Relative Minimum • A function value f (b) is a relative minimum of f if there exists an open interval containing b such that f (b) < f (x) for all x ≠ b in the open interval.

  34. HOW DO WE IDENTIFY EVEN AND ODD FUNCTIONS AND SYMMETRY? • Definition of Even and Odd Functions The function f is an even function if f (-x)= f (x) all x in the domain of f. The right side of the equation of an even function does not change if x is replaced with –x. The function f is an odd function if f (-x) = -f (x) for all x in the domain of f. Every term on the right side of the equation of an odd function changes its sign if x is replaced with –x.

  35. DETERMINE IF FUNCTION IS EVEN,ODD OR NEITHER • f (x) = x3 - 6x • Steps: • Replace x with –x and simplify. • If the right side of the equation stays the same it is an even function. • If every term on the right side changes sign, then the function is odd.

  36. DETERMINE IF FUNCTION IS EVEN, ODD OR NEITHER • g (x) = x4 - 2x2 • h(x) = x2 + 2 x + 1 • Steps: • Replace x with –x and simplify. • If the right side of the equation stays the same it is an even function. • If every term on the right side changes sign, then the function is odd.

  37. PRACTICE: • Determine if function is Even, Odd or Neither • f (x) = x2 + 6 • g(x) = 7x3 – x • h (x) = x5 + 1

  38. The function on the left is even. • What does that mean in terms of the graph of the function? • The graph is symmetric with respect to the y-axis. For every point (x, y) on the graph, the point (-x, y) is also on the graph. • All even functions have graphs with this kind of symmetry.

  39. The graph of function f (x) = x3 is odd. • It may not be symmetrical with respect to the y-axis. It does have symmetry in another way. • Can you identify how?

  40. For each point (x, y) there is a point (-x, -y) is also on the graph. • Ex. (2, 8) and (-2, -8) are on the graph. • The graph is symmetrical with respect to the origin. • All ODD functions have graphs with origin symmetry.

  41. SUMMARY:ANSWER IN COMPLETE SENTENCES. • What does it mean if a function f is increasing on an interval? • If you are given a function’s equation, how do you determine if the function is even, odd or neither? • Determine whether each function is even, odd or neither. a. f (x) = x2- x4 b. f (x) = x(1- x2)1/2

  42. AIM #1.3B: HOW DO WE UNDERSTAND AND USE PIECEWISE FUNCTIONS? • A piecewise function is a function that is defined by two (or more) equations over a specified domain.

  43. Example: ( DO NOTCOPY) READ • A cellular phone company offers the following plan: • $20 per month buys 60 minutes • Additional time costs $0.40 per minute

  44. HOW DO WE EVALUATE A PIECEWISE FUNCTION?

  45. PRACTICE: • Find and interpret each of the following: a. C (40) b. C (80)

  46. HOW DO WE GRAPH A PIECEWISE FUNCTION?

  47. We can use the graph of a piecewise function to find the range of f. • What would the range be for the piecewise function? ( For previous piecewise function)

  48. Some piecewise functions are called step functions because the graphs form discontinuous steps. • One such function is called the greatest integer function, symbolized by int (x) or • int (x) = greatest integer that is less than or equal to x. • For example: a. int (1) = 1, int (1.3) = 1, int (1.5) = 1, int (1.9)= 1 b. int (2) = 2, , int (2.3) =2 , int (2.5) = 2, int (2.9)= 2

  49. Graph of a Step Function

  50. FUNCTION AND DIFFERENCE QUOTIENTS • Definition of the Difference Quotient of a Function: The expression for h≠0 is called the difference quotient of the function f.

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