1 / 78

Ch 2 Quarter TEST Review

Ch 2 Quarter TEST Review. RELATION. A correspondence between 2 sets …say you have a set x and a set y, then… x corresponds to y y depends on x x is the input and y is the output x y. 0 1 -1. 1 3 -1. Ways to express a relation:. Equation

gsnyder
Télécharger la présentation

Ch 2 Quarter TEST Review

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Ch 2 Quarter TESTReview

  2. RELATION A correspondence between 2 sets …say you have a set x and a set y, then… x corresponds to y y depends on x x is the input and y is the output x y

  3. 0 1 -1 1 3 -1 Ways to express a relation: Equation Table Graph Mapping

  4. FUNCTION A relation from set x into set y that associates with element of x with exactly one element of y. • inputs MUST have only ONE output • outputs MAY be repeated for different inputs

  5. Example 1ATell if the relation is a function; if so state domain and range Mom Dad Kari Seth 555-2341 555-7890 555-8541 555-2222 555-3213 555-8504 NOT A FUNCTION!!!

  6. Example 1B Tell if each relation is a function; if so state domain and range Domain: {Hamburger, Cheeseburger, Chef Salad} Hamburger Cheese-burger Chef Salad $4.00 $4.50 $5.00 Range: {$4.00, $4.50, $5.00} YES a function!!!

  7. Example 1C Tell if each relation is a function; if so state domain and range Even though 23 occurs twice, just list it once in the range… Domain:{410, 580, 750, 600, 430} Range: {19, 29, 33, 23} YES a function!!!

  8. Example 1D Tell if each relation is a function; if so state domain and range {(1,4), (3,5), (5,7), (7,9)} YES a function!! Domain: {1,3,5,7} Range: {4,5,7,9}

  9. Example 1E Tell if each relation is a function; if so state domain and range {(4,-2), (1,-1), (0,0), (1,1), (4,2)} NOT a function!! Because the 4 and 1 in the domain each have more than one output!

  10. Example 1F Tell if each relation is a function; if so state domain and range {(-2,4), (-1,1), (0,0), (1,1), (2,4)} YES a function!! Outputs CAN be repeated…just not elements of the domain! Domain: {-2,-1,0,1,2} Range: {4,1,0}

  11. So, if we have a table of values or a mapping, we can tell whether or not our relation is a function. But, what about equations? (1) Solve for y (put the function in its explicit form) (2) Check if any value in the domain will generate more than one value for y. a) Look at the table, or b) Look at the graph (vertical line test)

  12. Example 2 Determine whether the equation defines y as a function of x. Example B Example A -x2 -x2 +3x +3x YES a function! Step 1: solve for y by subtracting x 2 from both sides… Step 1: solve for y by adding 3x to each side… NOT a function!! Take the square root: Step 2: graph on calculator/look at table… Step 2: graph the two equations on calculator…

  13. Finding Values of a Function • f(x) means “the value of f at the number x” • The independent variable is called the argument x is the input and f(x) is the output or value…

  14. Example 3: Evaluate the following for the function, f, defined by Just plug it in & simplify! A. f(3) = 3(3)2 + 5(3) =42 B. f(x) + f(3) Remember: f(3) = 42… = f(x) + 42 …and f(x) is 3x2 + 5x = 3x2 + 5x+ 42 = 3x2 + 5x + 42

  15. Example 3 cont… C. f(-x) = 3(-x)2 + 5(-x) (-x)2 =x 2 = 3x 2 + 5(-x) 5(-x) = -5x = 3x2 - 5x D. - f(x) This means the opposite of f(x)… = - (3x 2+ 5x) …distribute the negative… = - (3x 2+ 5x) = -3x2 – 5x

  16. Example 3 cont… E. f(x + 3) = 3(x + 3)2 + 5(x + 3) (x + 3) 2 = (x + 3)(x + 3) = x2 +6x + 9 = 3(x2 + 6x + 9) + 5(x + 3) distribute the 3 and 5… + 18x + 15 = 3x 2 + 5x + 27 …combine like terms = 3x 2 + 23x + 42

  17. Example 3 cont… F. 1st take care of f(x + h) f(x + h)= 3(x + h) 2 + 5(x + h) = 3(x 2 + 2xh + h 2) + 5(x + h) distribute the 3 and 5… + 6xh + 3h 2 + 5x + 5h = 3x 2 f(x + h) = 3x 2 + 6xh + 3h 2 + 5x + 5h

  18. Example 3F cont… f(x + h) – f(x) = = 3x 2 + 6xh + 3h 2 + 5x + 5h - (3x 2 + 5x) distribute the negative… - 5x = 3x 2 + 6xh + 3h 2 + 5x + 5h - 3x 2 …combine like terms… …and factor. = h (6x + 3h + 5) = 6xh + 3h 2 + 5h = 6x + 3h + 5

  19. The Domain of a Function Huh?? If an equation for a function, f, is given with no specified domain, it is agreedthat the domain of f is the largest set of real numbers for which the value of f(x) is a real number. In other words: The domain is all inputs that make sense and give an answer for the equation…

  20. Example 4: Find the domain of each function. What can x NOT be?? Verify with a graph! x can be anything! A. all real numbers Cannot have a zero on the bottom! B. x is undefined at -1 and 1

  21. Example 4 continued: Find the domain of each function. You cannot take the square root of a negative number! C. +5t …add 5t to both sides… +5t …divide both sides by 5… or

  22. Functions can be added, subtracted, multiplied and divided: Operations on Functions If f and g are functions, then...

  23. Example 5A: Given f(x) =3x + 7 and g(x) = 2x – 4, find f + g, f - g, f * g, and f /g. f+g: (3x + 7) + (2x - 4) = 5x+3 f-g:(3x + 7) - (2x - 4) = 3x + 7 – 2x + 4 = x+11 (combine like terms) (distribute negative sign) (combine like terms)

  24. f *g: (3x + 7)(2x - 4) Example 5A continued: Given f(x) =3x + 7 and g(x) = 2x – 4 = 3x + 7 2x - 4 (FOIL...multiply) – 12x + 14x – 28 = 6x 2 (simplify) = 6x2 + 2x - 28 f/g: (3x + 7)/(2x – 4) (There is nothing more to do!)

  25. f+g: Example 5B: Given f(x) =3/x and g(x) = 1/x, find f + g, f - g, f * g, and f /g. (must have a common denominator whenever adding or subtracting fractions) f-g:

  26. f *g: Example 5B continued: Given f(x) =3/x and g(x) = 1/x (top x top bottom x bottom) f/g: (flip and multiply!)

  27. Vertical Line Test A set of points in the xy-plane is the graph of a function iff (if and only if) every vertical line intersects the graph in at most one point. Example: Determine whether the graph represents a function. Yes No Yes No

  28. Many things can be learned about a function from its graph, such as (but certainly not limited to) values of the function (f(x)) at various values of x, the domain and range, the intercepts (both x and y), the number of times a function is intersected by other functions, and values of x that generate different function values.

  29. EXAMPLE • Is the point (1, ½) on the graph of f? (use table on calc.) yes • If x = 2, what is f(x)? What point is on the graph of f? f(x)=⅔ (2, ⅔) • If f(x) = 2, what is x? What point is on the graph of f? x = -2 (-2,2) • Find any x-intercepts. (0, 0) • Find any y-intercepts. (0, 0) (e)

  30. EXAMPLE The average cost C of manufacturing x computers per day is given by the function Determine the average cost of manufacturing: (a) 30 computers in a day (b) 40 computers in a day (c) 50 computers in a day $1351.54 $1232.97 $1293.07

  31. What do the y-values need to be?

  32. What value of x minimizes the average cost? Use TRACE to find approximate minimum, then TABLE to verify Producing 41 computers a day will minimize the cost to $1231.75.

  33. Determining Even and Odd Functions from the Graph A function, f, is even if for every number x in its domain the number –x is also in the domain and f(-x) = f(x)...that is, if (x, y) is on the graph, then (-x, y) is too. Also, a function is even iff its graph is symmetric with respect to the y-axis. f (-x) = f (x)

  34. An EVEN Function f (-x) = f (x) (-2, 2) & (2, 2) Graph must go in same direction and be symmetric to the y-axis!

  35. Determining Even and Odd Functions from the Graph A function, f, is odd if for every number x in its domain the number –x is also in the domain and f(-x) = -f(x)...that is, if (x, y) is on the graph, then (-x, -y) is too. Also, a function is odd iff its graph is symmetric with respect to the origin. f (-x) = -f (x)

  36. An ODD Function f (-x) = -f (x) (-4, 8) & (4, -8) Graph must go in opposite directions and intersect the origin!

  37. Example: Determine whether each graph given is an even function, an odd function, or neither even nor odd. Even Neither Odd

  38. Identifying Even and Odd Functions Algebraically EXAMPLE Determine whether each function is even or odd...verify algebraically. Graph it…does it appear to be even or odd? A. To verify algebraically, you must prove that f(-x) = f(x) Since f(-x) = f(x), then it’s an EVEN function!

  39. Identifying Even and Odd Functions Algebraically B. Graph it…does it appear to be even or odd? To verify algebraically, first prove that So now prove that Therefore it is not even… Not odd… Then it’s NEITHER even nor odd!

  40. Identifying Even and Odd Functions Algebraically Graph it…does it appear to be even or odd? C. First find f(-x)... Then find -f(x)... Since f(-x) = -f(x), then it’s an ODD function!

  41. Determining Where a Function is Increasing, Decreasing, or Constant Look from left to rightalong a graph and find the parts of the graph that are rising (increasing), falling (decreasing), and horizontal (constant). REMEMBER: these are INTERVALS named by the x coordinates!

  42. Example Where is the function decreasing? (-6, -4) U (3, 6) Where is the function increasing? (-4, 0) Where is the function constant? (0, 3)

  43. Finding Local Maximums and Local Minimums Maximums are the hills and minimums are the valleys... REMEMBER: these are named by the y coordinates! On your calculator: TRACE to the point, then 2nd – calc – (3) minimum or (4) maximum

  44. Example What are the maximums, if any? A max is where y = 2. What are the minimums, if any? There are 2 mins: one at y = 1 and another at y = 0. List the intervals on which f is increasing and decreasing. U (1, 3) Decreasing: (-1, 1) Increasing:

  45. Example Use the graphing calculator to graph the function for . Find local maximums and minimums and determine the intervals of increase and decrease. NOTE: Always sketch your graph and fill in the points!

  46. Max (-.707, 2.414) Min (.707, -0.414)

  47. (2, 11) (-0.707, 2.414) (0.707, -0.414) (-2, -0.707) U (0.707, 2) Increasing: (-0.707, 0.707) Decreasing: (-2, -9)

  48. Average Rate of Change of a Function If c is in the domain of a function, f, the average rate of change of f from c to x is defined: , where This is just calculating the slope between two points!! (In calculus, this expression is called the difference quotient of f at c.)

  49. Example Find the average rate of change of f(x) = x2 – 2x + 3 from… (-1, 6) and (1, 2) ...-1 to 1: When x = -1, y = 6 When x = 1, y = 2 ...0 to 2: (0, 3) and (2, 3) When x = 0, y = 3 When x = 2, y = 3

  50. 3 -1 rise = run Graph of a Linear Function -1 = -3 • y = mx + b • It has a constant slope, m • The y-intercept is b • y and x are the variables, representing vertical and horizontal changes, respectively 3 m = -3 b = 7 y = -3x + 7

More Related