Chapter 3 Transformations of Graphs and Data

Transcription

1 Chapter 3 Transformations of Graphs and Data 3.1 Graphs of Parent Functions Parent Function the simplest equation of a particular type of function o Ex: Quadratic Function: y = x 2 There are EIGHT important parent functions: Choosing a window while graphing functions on a calculator The window allows you to zoom in or zoom out to see your graphs differently It also allows you to view graphs that have been shifted left, right, up or down on the coordinate plane You always want to choose a window that is going to allow you to see the most important aspects of each function!!!!! Ex 1: a. Graph g(x) = x on your graphing calculators. Determine an appropriate window that allows you to see what the graph looks like. b. What do you notice about how the graph is being shifted? c. What is the domain and range of the function g?

2 Ex 2: a. Graph the real function g with the equation g(x) = x on your graphing calculators. Determine an appropriate window that allows you to see the important features of the graph. b. What do you notice about how the graph is being shifted? c. State the domain and range of the function g. Ex 3: Stephen, who is 1.7 m tall, dives off a 3 m springboard. An equation modeling his height h(x) in meters above the water at time x in seconds is h(x) = 4.9x x a. Create a graph that would be helpful for determining the maximum height of his dive and how long it lasts. State the window that would be appropriate. b. What is the domain and range of h within the context of this situation? Asymptotes Consider the graph f(x) = 1. Notice that the asymptote occurs when x = 5 this is because x 5 and division by is Remember that when you are graphing functions with asymptotes on a graphing calculator there is not dotted line to represent the discontinuity in the graph.

3 Ex 4: Graph y = 1 3x+2 by hand. Be sure to label your axes and graph your asymptote. 3.2 The Graph-Translation Theorem Transformation a one-to-one correspondence between sets of points o Ex: Preimage the set of points before the transformation Image the mapping of a set of points pot transformation Translation a shifting of the graph horizontally, vertically or both. o Ex: Translation A translation in the plane is a transformation that maps each point (x, y) onto (x + h, y + k), where h and k are constant. Up = Down = Right = Left = Ex: Write a rule for the above translation

4 Ex 1: The graph of y = x 2 is shown to the right, together with its image under a translation T. The point (0, 0), which is the vertex of the preimage, maps onto the vertex (-3, 1) of the image. a. Find a rule for the translation T b. Find the image of (2, 4) under this translation. Ex 2: The graph of a triangle and its image under translation T is the to right. a. Find a rule for the translation T b. Find the image of (3, -1) under this translation The Graph-Translation Theorem The graph below shows a circle in BLUE with equation It is being translated units and units The graph of the translated circle has equation The rule for the translation is What do you notice about the relationship between the equation of the image and the rule of the translation? Can you give me an example of a different graph that has a specific translation and how it would effect the equation of the graph?

5 Ex 3: At the right are graphs of the function y = C(x) = 25 x 2 and its image y = D(x) under the translation (x, y) (x + 5, y 4). Both are semicircles. Find an equation for the image. Ex 3: Use the graph of y = 1 1 x2 to sketch the graph of y = 4 (x+1) 2

6 3.3 Translations of Data Centers of Translated Data Adding h to each number in a data set to each of the mean, median and mode. Spread of Translated Data Adding h to each number in a data set the range, interquartile range, variance, or standard deviation of the data. Because the measures of spread of a data set under a translation, they are said to be under a translation Ex 1: In a local produce store, cantaloupes sell for 99 cents per pound. A clerk weighed 30 melons and computed the following statistics: Mean = 3.5 lbs., Standard Deviation = 8 oz., median = 3.4 lbs., and IQR = 1 lb. After finishing his task, the clerk noticed that the scales were not correctly calibrated. The scale was set at 3 oz. as its starting weight, not 0. Fins the correct values for the mean, standard deviation, median and IQR for the 30 melons. 3.4 Symmetries of Graphs Reflection-symmetric figure when a figure can be mapped onto itself by a reflection over some o Ex:

7 Axis/Line of Symmtery the reflecting line l Point Symmetry when a figure can be mapped onto itself under a rotation of 180 around o Ex: Center of Symmetry the point at which the figure is being rotated about Ex 1: The diagram at the right shows half of a graph. a. Copy the diagram. Draw the other half of the graph so that the result is point-symmetric about the origin. Label this half A b. Draw the other half of the original graph so that the result is symmetric with respect to the y- axis. Label this half B c. Draw the other half of the original graph so that it is symmetric over the x-axis. Label the graph C. d. What symmetries does the union of graphs A, B and C and the original graph possess? Symmetry over y-axis Symmetry over x-axis Symmetry about the origin

8 Proving Symmetry Ex 2: Prove that the graph of f(x) = 1 x 2 +1 Even and Odd Functions Any function whose graph is symmetric with respect to the is called an o A function is iff for all values of x in its domain Any function whose graph is symmetric about the is called an o A function is iff for all values of x in its domain Any other function is said to be Ex 3: Use your graphing calculator to graph the function f(x) = 4x 2x 3. Determine whether the function is odd, even or neither. If it appears to be odd or even, prove it. Ex 4: Graph the function F with y = f(x) = 1 (x+3) 7 a. Give equations for the asymptotes of the graph b. Describe any lines or points of symmetry

9 3.5 The Graph Scale-Change Theorem The graph of a function can be scaled, or Activity 1! Use DESMOS to do the following: a. Graph f(x) = x 3 + 3x 2 4x with window 11 x 13 and 10 y 40 b. Graph g(x) = 3(x 3 + 3x 2 4x) = 3 f(x) on the same axes. Fill in the table of values for f(x) and g(x) only x f(x) g(x) 0.5 f(x) 1.5 f(x) 2 f(x) c. Repeat step 2 with other values of f(x) = b(x 3 + 3x 2 4x) and use a slider to vary the value of b. Complete the above chart by setting the slider to 0.5, 1.5 and then 2. Describe how these values related to the original values of f(x) Activity 2! Use Desmos to do the following: a. Consider the graph of f(x) = x 3 + 3x 2 4x from Activity 1. Complete the table. b. If h(x) = f( x ) then h(x) = a (x a )3 + 3( x a )2 4( x ). Graph h(x) and use a a slider to vary the value of a c. What are the x- and y- intercepts of f(x) x f(x) d. How do the intercepts of h(x) change as a changes.

10 The Graph Scale-Change Theorem Scale change a transformation that maps, where and are constants o Vertical scale change o Horizontal scale change o Size change Example: Given a primage graph described by a sentence in x and y, the following two processes yield the same image graph: Ex 1: The relation described by x 2 + y 2 = 25 is graphed at the right a. Find images of points labeled A-F on the graph under S: (x, y) (2x, y). b. Copy the circle onto the graph and then graph the image on the same axes. c. Write an equation for the image relation.

11 3.6 Scale Changes of Data Ex 1: Suppose a refrigerator cost $800 in What might you expect the cost of a similar refrigerator to be in June Ex 2: As a fund-raiser, club members sell candy for $2.50 per box. The number of boxes each of the 17 members sold is given below. a. Compute the mean, median, standard deviation, and IQR of the numbers of boxes sold. b. Find the amount of money each member collected. c. Compute the mean, median, standard deviation and IQR of the amount of money collected by scaling the values in part a. Multiplying every number in a data set by k multiplies all measures of and the and by, while the is multiplied by

12 3.7 Composition of Functions Functions can be used to describe relationships between members of this tree Composition is not Commutative Definition of Composite Functions Suppose f and g are functions. The composite of g with f, written is the function defined by: The domain of is the set of values of in the domain of for which is in the domain of. Ex 1: Let f and g be defined by f(x) = 2x 2 + 3x and g(x) = x 7. Evaluate. a. (f g)( 2) b. (g f)( 2)

13 Ex 2: Let f(x) = 2x 2 + 3x and g(x) = x 7. a. Derive a formula for (f g)(x) b. Derive a formula for (g f)(x) Ex 3: Let f and g be real functions defined by f(m) = m and g(m) = 2 m 3. Composition of Transformations Ex 4: Let S: (x, y) (2x, y) and let T: (x, y) (x + 4, y 3) a. Describe S and T in words b. Write a formula for the composite (T S)(x, y) and describe it in words c. Write a formula for the composite (S T)(x, y) and describe it in words

14 3.8 Inverses of Functions Remember that a function can be considered as a set of in which each first element is paired with exactly one second element. o The is found by switching the coordinates of the pairs Ex 1: Let f = {(-3, -5), (-2, 0), (-1, 3), (0, 4), (1, 3), (2, 0), (3, -5)}. Describe the inverse of f. Is the inverse a function? If the original function is described by an then in the equation gives an equation its. Ex 2: a. Give an equation for the inverse of the function described by y = x b. Sketch a graph of y = x and the inverse on the same set of axes c. Is the inverse a function?

15 Ex 3: a. Give an equation for the inverse of the function described by f(x) = 1 x b. Graph f and its inverse on the same axes Inverse Functions Theorem Given any two functions f and g, f and f are inverse functions iff for all x in the domain of g, and for all x in the domain of f. o Identity function Ex 4: Use the inverse functions theorem to determine whether f and g are, defined by f(x) = 2x 4 x 1 and g(x) = x+1 2x+4 Ex 5: Use the inverse functions theorem to determine whether f and g are, defined by f(x) = 3x 2 and g(x) = x+2 3

16 3.9 Z-Scores Sometimes a person wants to know how his or her score or salary compares to a group as a whole o One way is to analyze how many the score or salary is or the mean. Z-Score Suppose a data set has mean and standard deviation. The z-score for a member x of this data set is A z-score tells how many standard deviations the score is the mean. A z-score tells how many standard deviations the score is the mean. Ex 1: Nancy scored 87 on a math quiz on which the mean score was 70 and the standard deviation of the scores was 8. Find her z-score and tell how far her score was from the mean. Original data = or Data resulting from transformations Properties of Z-scores If a data set has mean and standard deviation, the mean of its z-scores (if each score is converted to a z- score) will be and the standard deviation of its z-scored will be Ex 2: Mark scored 78 a history test on which the mean was 71 and the standard deviation was 10. He scored 68 on a chemistry test on which the mean was 62 and the standard deviation was 6. Use z-scores to determine on which test he performed better compared to his classmates.