Math 2 Spring Unit 5 Bundle Transformational Graphing and Inverse Variation

Transcription

1 Math 2 Spring 2017 Unit 5 Bundle Transformational Graphing and Inverse Variation 1

2 Contents Transformations of Functions Day Transformations with Functions Day 1 HW Transformations with Functions Day Transformations with Functions Day 2 HW Transformations with Functions Day Transformations with Functions Day 3 HW Transformations of Quadratic Functions Day Transformations of Quadratic Functions Day 4 HW Transformations of Radical Functions Day Transformations of Radical Functions Day 5 HW Direct and Inverse Variation Day Direct and Inverse Variation Day 5 HW Graphing Inverse Variation Day Graphing Inverse Variation Day 6 HW Unit 5 Review

3 Transformations of Functions Day 1 To the right is a graph of a function F(x). We can use F(x) functions to explore transformations in the coordinate plane. I. Let s review briefly. 1. a. Explain what a function is in your own words. b. Using the graph, how do we know that F(x) is a function? F(x) 2. a. Explain what we mean by the term domain. b. Using the graph, what is the domain of F(x)? 3. a. Explain what we mean by the term range. b. Using the graph, what is the range of F(x)? 4. Let s explore the points on F(x). a. How many points lie on F(x)? Can you list them all? b. What are the key points that would help us graph F(x)? We are going to call these key points characteristic points. It is important when graphing a function that you are able to identify these characteristic points. c. Use the graph of graph to evaluate the following. F(1) = F( 1) = F( ) = 2 F(5) = 3

4 II. Remember that F(x) is another name for the y-values. Therefore the equation of F(x) is y = F(x). x 1 F(x) Why did we choose those x-values to put in the table? Now let s try graphing G(x): y = F(x) + 4. Complete the table below for this new function and then graph G(x)on the coordinate plane above. y = F(x) + 4 x y What type of transformation maps F(x), to G(x), F(x) + 4? (Be specific.) 3. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 4. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 5. In y = F(x) + 4, how did the +4 affect the graph of F(x)? Did it affect the domain or the range? 4

5 III. Suppose G(x) s equation is: y = F(x) 3. Complete the table below for this new function and then graph G(x)on the coordinate plane above. y = F(x) 3 x y What type of transformation maps F(x), F(x), to G(x), F(x) 3? Be specific. 2. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 3. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 4. In y = F(x) 3, how did the 3 affect the graph of F(x)? Did it affect the domain or the range? IV. Checkpoint: Using the understanding you have gained so far, describe the affect to F(x) for the following functions. Equation Example: y=f(x) + 18 Effect to F(x) s graph Translate up 18 units 1. y = F(x) y = F(x) y = F(x) y = F(x) 521 5

6 V. Suppose G(x) s equation is: y = F(x + 4). 1. Complete the table. x x + 4 y (Hint: Since, x + 4 = 1, subtract 4 from both sides of the equation, and x = 5. Use a similar method to find the missing x values.) 2. On the coordinate plane above, graph the 4 ordered pairs (x, y). The first point is ( 5, 1). 3. What type of transformation maps F(x), F(x), to G(x), F(x + 4)? (Be specific.) 4. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 5. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 6. In y = F(x + 4), how did the +4 affect the graph of F(x)? Did it affect the domain or the range? 6

7 VI. Suppose G(x) s equation is: y = F(x 3). Complete the table below for this new function and then graph G(x)on the coordinate plane above. 1. Complete the table. y = F(x 3) x x 3 y On the coordinate plane above, graph the 4 ordered pairs (x, y). [Hint: The 1 st point should be (2, 1).] 3. What type of transformation maps F(x), F(x), to G(x), F(x 3)? (Be specific.) 4. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 5. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 6. In y = F(x 3), how did the 3 affect the graph of F(x)? Did it affect the domain or the range? 7

8 VII. Checkpoint: Using the understanding you have gained so far, describe the effect to F(x) for the following functions. Equation Example: y=f(x + 18) Effect to F(x) s graph Translate left 18 units 1. y = F(x 10) 2. y = F(x) y = F(x + 48) 4. y = F(x) y = F(x + 30) + 18 VIII. Checkpoint: Using the understanding you have gained so far, write the equation that would have the following effect on F(x) s graph. Equation Example: y=f(x + 8) 1. Effect to F(x) s graph Translate left 8 units Translate up 29 units 2. Translate right 7 3. Translate left Translate left 5 and up Translate down 2 and right 6 8

9 IX. Now let s look at a new function. Its notation is H(x). Use H(x) to demonstrate what you have learned so far about the transformations of functions. 1. What are H(x) s characteristic points? 2. Describe the effect on H(x) s graph for each of the following. a. H(x 2) b. H(x) + 7 c. H(x+2) 3 3. Use your answers to questions 1 and 2 to help you sketch each graph without using a table. a. y = H(x 2) b. y = H(x) + 7 c. y = H(x+2) 3 9

10 Transformations with Functions Day 1 HW I. On each grid, R(x) is graphed. Graph the given function. 1. Graph: y = R(x) Graph: y = R(x + 2) Graph: y = R(x + 6) 4. Graph: y = G(x 4) 5 10

11 II. Using the understanding you have gained so far, describe the effect to F(x) for the following functions. Equation Effect to F(x) graph 6. y = F(x) y = F(x 13) 8. y = F(x + 9) 9. y = F(x) y = F(x 25) + 11 III. Using the understanding you have gained so far, write the equation that would have the following effect on F(x) s graph. Equation Effect to F(x) s graph 6. Translate left 51 units 7. Translate down Translate right Translate right 8 and down Translate down 12 and left 100 IV. Determine the domain and range of each parent function. 1. H(x) 2. R(x) Domain: Domain: Range: Range: 11

12 V. Consider a new function, P(x). P(x) s Domain is {x 2 x 2}. Its range is {y 3 y 1}. Use your understanding of transformations of functions to determine the domain and range of each of the following functions. (Hint: You may want to write the effect to P(x) first.) 1. P(x) P(x + 5) Domain: Domain: Range: Range: 12

13 Transformations with Functions Day 2 Today we will revisit F(x), our parent function, and investigate transformations other than translations. Recall that the equation for F(x) is y = F(x). Complete the chart with F(x) s characteristic points. x F(x) F(x) I. Let s suppose that G(x) is y = F(x) 7. Complete the table. y = F(x) x F(x) y On the coordinate plane above, graph the 4 ordered pairs (x, y). [Hint: The 1 st point should be ( 1, 1).] 9. What type of transformation maps F(x) to G(x), F(x)? (Be specific.) 10. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 11. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 12. In y = F(x), how did the negative coefficient of F(x) affect the graph of F(x)? How does this relate to our study of transformations earlier this semester? 13

14 II. Now let s suppose that G(x) is y = F( x) 1. Complete the table. y = F( x) x x y 1 F(x) On the coordinate plane above, graph the 4 ordered pairs (x, y). [Hint: The 1 st point should be (1, 1).] 3. What type of transformation maps F(x) to G(x), F( x)? (Be specific.) 4. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 5. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 6. In y = F( x), how did the negative coefficient of x affect the graph of F(x)? How does this relate to our study of transformations earlier this semester? III. Checkpoint: H(x) is shown on each grid. Use H(x) s characteristic points to graph H(x) s children without making a table. 1. y = H( x) 2. y = H(x) 14

15 IV. Now let s return to F(x), whose equation is y = F(x). Complete the chart with F(x) s characteristic points. x F(x) F(x) Let s suppose that G(x) is y = 4 F(x) 1. Complete the table. y = 4 F(x) x F(x) y On the coordinate plane above, graph the 4 ordered pairs (x, y). [Hint: The 1 st one should be ( 1, 4).] 3. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 4. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 5. In y = 4 F(x), the coefficient of F(x) is 4. How did that affect the graph of F(x)? Is this one of the transformations we studied? If so, which one? If not, explain. 15

16 V. Now let s suppose that G(x) is y = ½ F(x). 1. Complete the table. y = ½ F(x) x F(x) y 1 F(x) On the coordinate plane above, graph the 4 ordered pairs (x, y). [Hint: The 1 st one should be ( 1, ½).] 3. How did this transformation affect the x-values? (Hint: Compare the characteristic points of F(x) and G(x)) 4. How did this transformation affect the y-values? (Hint: Compare the characteristic points of F(x) and G(x)) 5. In y = ½ F(x), the coefficient of F(x) is ½. How did that affect the graph of F(x)? How is this different from the graph of y = 4 F(x) on the previous page? VI. Checkpoint: 1. Complete each chart below. Each chart starts with the characteristic points of F(x). x F(x) 3 F(x) x F(x) ¼ F(x) Compare the 2 nd and 3 rd columns of each chart above. The 2 nd column is the y-value for F(x). Can you make a conjecture about how a coefficient changes the parent graph? 16

17 VII. Now let s suppose that G(x) is y = 3 F(x). 1. Complete the table. y = 3 F(x) x F(x) y 1 F(x) On the coordinate plane above, graph the 4 ordered pairs (x, y). [Hint: The 1 st one should be ( 1, 3).] 3. Reread the conjecture you made in #7 on the previous page. Does it hold true or do you need to refine it? If it does need some work, restate it more correctly here. VIII. Checkpoint: Let s revisit H(x). 4. Describe the effect on H(x) s graph for each of the following. Example: 5H(x) Each point is reflected in the x-axis and is 5 times as far from the x-axis. d. 3H(x) e. 2H(x) f. 1 2 H(x) 5. Use your answers to questions 1 and 2 to help you sketch each graph without using a table. b. y = 3H(x) b. y = 2H(x) c. y = 1 2 H(x) 17

18 Transformations with Functions Day 2 HW This is the function B(x). 1. List its characteristic points. 2. Are these the only points on the graph of B(x)? Explain. 3. What is the domain of B(x)? 4. What is the range of B(x)? For each of the following, list the effect on the graph of B(x) and then graph the new function. 5. y = B( x) 6. y = B(x) 7. y = 1 3 B(x) 8. y = 3 B(x) 9. y = B(x 3) 10. y = B(x + 2) 1 18

19 Transformations with Functions Day 3 The graph of D(x), is shown. List the characteristic points of D(x). What is different about D(x) from the functions we have used so far?. Since D(x) is our original function, we will refer to him as the parent function. Using our knowledge of transformational functions, let s practice finding children of this parent. Note: In transformational graphing where there are multiple steps, it is important to perform the translations last. I. Example: Let s explore the steps to graph D(x) Jr, 2D(x + 3) + 5, without using tables. Step 1. The transformations represented in this new function are listed below in the order they will be performed. (See note above.) Vertical stretch by 2 (Each point moves twice as far from the x-axis.) Translate left 3. Translate up 5. Step 2. On the graph, put your pencil on the left-most characteristic point, ( 5, 1). Vertical stretch by 2 takes it to ( 5, 2). (Note that the originally, the point was 1 unit away from the x-axis. Now, the new point is 2 units away from the x-axis.) Starting with your pencil at ( 5, 2), translate this point 3 units to the left. Your pencil should now be on ( 8, 2). Starting with your pencil at ( 8, 2), translate this point up 5 units. Your pencil should now be on ( 8, 3). Plot the point ( 8, 3). It is recommended that you do this using a different colored pencil. Step 3. Follow the process used in Step 2 above to perform all the transformations on the other 3 characteristic points. Step 4. After completing Step 3, you will have all four characteristic points for E(x) Use these to complete the graph of E(x) Be sure you use a curve in the appropriate place. D(x) is not made of segments only. 19

20 II. D(x) has another child named C(x), D(x) 4 Using the process in the previous example as a guide, graph C(x) (without using tables). 1. List the transformations needed to graph C(x). (Remember, to be careful with order.). 2. Apply the transformations listed above to each of the four characteristic points. 3. Complete the graph of C(x) using your new characteristic points from #2. III. D(x) has another child named A(x), 3 D( x) Using the process in the previous example as a guide, graph A(x) (without using tables). 1. List the transformations needed to graph A(x). (Remember, to be careful with order.). 2. Apply the transformations listed above to each of the four characteristic points. 3. Complete the graph of A(x) using your new characteristic points from #2. 20

21 IV. Now that we have practiced transformational graphing with D(x) and his children, you and your partner should use the process learned from the previous three problems to complete the following. 1. Given K(x), graph: y = 3K(x) Given J(x), graph: y = J(x 3) 6 3. Given H(x), graph y = 3H(x) 21

22 4. Given B(x), graph: y = B( x) Given M(x), graph: y = 1 3 M(x) 22

23 V. Finally, let s examine a reflection of H(x) in the line y = x. 1. Graph this line (y = x) on the grid. 2. Using H(x) s characteristic points and the MIRA, graph H(x) s reflection. 3. Complete the charts below with the characteristic points: y = H(x) H(x) s reflection in y = x: x y x y 4. Compare the points in the two charts. Describe what happens when we reflect in the line y = x. (This should match what we learned in our earlier study of reflections in the line y = x.) 5. A reflection in the line y = x, shows a graph s inverse. We will study this in more depth in a future unit. Look at the graph of H(x) s inverse. Is the inverse a function? Explain how you know. 23

24 Transformations with Functions Day 3 HW 1. List the transformations needed to graph the following. Remember that translations are done last. a. y = 2F(x) + 2 b. y = 1 F(x 6) 3 c. y = F(x) 12 d. y = 3F(-x) e. y = 5F(x) 2. Looking back at the examples of parent functions we have worked with, create your own original parent function on the graph. Make sure that you have graphed a function. a. How can you tell your graph is a function? b. Explain the name you picked. c. Write an equation for your function that will have the following effects. Stretch vertically by 2 and translate left 4. Reflect in the x-axis and compress vertically by 1 2 Translate up 6 and right 4 d. Graph each of the children from part c above using a separate graph for each. You will need to use your own graph paper. 24

25 Transformations of Quadratic Functions Day 4 Quadratic Functions The parent function for a quadratic is The is NOT a transformation Example 1: Identify the transformations y = (x 3) You Try 1: Identify the transformations y = 3(x + 1) 2 3 You Try 2: Identify the transformations y = 2(x) For a quadratic function the critical points are: x y Domain: Range: Example 2: Find and graph the new critical points for the equation y = (x 3) a. Determine the transformations that affect the x-values x y b. Determine the transformations that affect the y-values a. Graph the new points New Domain: New Range: 25

26 Example 3: Find and graph the new critical points for the equation y = 3(x + 1) 2 3 a. Determine the transformations that affect the x-values x y b. Determine the transformations that affect the y-values a. Graph the new points New Domain: New Range: You Try 3: Find and graph the new critical points for the equation y = 2(x) a. Determine the transformations that affect the x-values x y b. Determine the transformations that affect the y-values b. Graph the new points New Domain: New Range: 26

27 Example 4: Write an equation for a quadratic function that is shifted right 6 and up 4 You Try 4: Write an equation for an quadratic function that is reflected over the x-axis, a vertical stretch of 2, and shifted up 6 27

28 Transformations of Quadratic Functions Day 4 HW 28

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30 Transformations of Radical Functions Day 5 Radical Functions The parent function for a radical function is Example 5: Identify the transformations y = x You Try 5: Identify the transformations y = 2 x + 4 You Try 6: Identify the transformations y = x 3 For a radical function the critical points are: x y Domain: Range: 30

31 Example 6: Find and graph the new critical points for the equation y = x x y c. Determine the transformations that affect the x-values d. Determine the transformations that affect the y-values b. Graph the new points New Domain: New Range: 31

32 Example 7: Find and graph the new critical points for the equation y = 2 x + 4 c. Determine the transformations that affect the x-values x y d. Determine the transformations that affect the y-values c. Graph the new points New Domain: New Range: 32

33 You Try 7: Find and graph the new critical points for the equation y = x 3 c. Determine the transformations that affect the x-values x y d. Determine the transformations that affect the y-values d. Graph the new points New Domain: New Range: Example 8: Write an equation for a radical function that is shifted down 6 and left 4 You Try 8: Write an equation for an radical function that is reflected over the x-axis, a vertical compression of 1/3, and shifted right 6 33

34 Transformations of Radical Functions Day 5 HW 34

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36 Direct and Inverse Variation Day 5 36

37 Examples: 1. The graph of a direct variation equation contains the point (3, -18). Find the constant of variation, k, and write the equation of the direct variation. 2. If y varies directly with x and y = 15 when x = -3, find x when y = The graph of a inverse variation equation contains the point (7, 10). Find the constant of variation, k, and write the equation of the inverse variation. 4. The graph of a inverse variation equation contains the point (5, 12). Find the constant of variation, k, and write the equation of the inverse variation. 5. Y varies inversely with x. If y = 21 when x = 10, find y when x = Y varies inversely with x. If y = 7.2 when x = 2.1, find x when y = The price for favors at the Sanderson High School Prom varies inversely with the number of students attending the prom. If 180 students attend, the cost of the party favors will be $10 each. What would be the price for each party favor if 240 students attend the prom? 37

38 Direct and Inverse Variation Day 5 HW Solve for the constant of variation (k) and write the inverse variation equation for the given points. 1. (3, 9) 2. (-9, -3) 3. ( 2 ) 4. (4.6, -3.5) 3, For a CD, the distance that a point on the disk is from the center varies inversely with its speed. If a point 2 inches from the center is traveling at 10 feet per second, how fast is a point 5.4 inches from the center traveling? 6. y varies directly with x. If y = 15 when x = -18, find y when x = y varies directly with x. If y = 75 when x =25, find x when y = 25. For the given points, find the inverse variation equation and find the missing coordinates of the second point. 8. (3, -5) (4, y) 9. (1.2, 3.6) (x, 2) 10. ( 2 ) (x, 8) 11. (0.5, 3) (9.2, y), The electric current I, is amperes, in a circuit varies directly as the voltage V. When 12 volts are applied, the current is 4 amperes. What is the current when 18 volts are applied? 13. When you are bulk ordering an item, the cost per item varies inversely with the number of items you are buying. If you order 300 pairs of sunglasses, they will cost 49 cents each. How many would you have to order for each pair to cost 25 cents? 14. The cost per ticket for a concert varies inversely with your distance to the stage. For a ticket in the first section, it costs $250 each. How much will it cost for a ticket in the 5 th section? If you are not going to spend more than $16 per ticket, what section would you find your seats? 38

39 Graphing Inverse Variation Day 6 39

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41 In your own words, define the following term. You may need to search for the terms online as a starting point. Asymptote: The inverse variation function has the standard form equation of y = 1. Fill in the table below using the pre-provided x- x values. x y Why do you think that zero is not used as an x value? Graph the order pairs on the provided graph Asymptotes are at and Because inverse variation is still a function, it can still be graphed using the transformations that we have done earlier in the unit. It is still possible to graph using translations up, down, left, and/or right, reflections over the x-axis and/or the y-axis, and vertical stretches or shrinks. 41

42 Example 1: Identify the transformation(s) in each of the following inverse variation functions. a) y = 1 x + 6 b) y = 1 x+6 c) y = 6 x d) y = 1 x You Try 1: Complete the table below by filling in the appropriate transformations Functions: f(x) = 5 x y = 3 x 6 8 f(x) = 2 x Transformations: Example 2: Graph the function y = 1 x List the transformations: New Asymptotes: and. X Y 42

43 Example 3: Graph the function y = 2 x 4 List the transformations: New Asymptotes: and. X Y You Try 2: Graph the function y = 3 x List the transformations: 43

44 Graphing Inverse Variation Day 6 HW For each of the functions below, list the transformations, give the domain and range, and graph the given function. 1. y = 1 x Transformations: Asymptotes 2. y = 2 x 3 6 Transformations: Asymptotes 3. y = 1 x+5 3 Transformations: Asymptotes 44

45 Unit 5 Review Use the graph on the right for the following problems: 1. What is the domain of the function? 2. What is the range of the function? F(x) 3. F(1) = 4. F( ) = 4 5. F(9) = List the critical points that are used for each of the following parent functions: 6. y = x 2 8. y = x 7. y = 1 x 9. y = 2 x y = 2(x + 3) 2 Transformations: Transformations: Domain: Range: Domain: Range: 45

46 11. y = 2 x 3 4 Transformations: Asymptotes: Domain: Range: Write the equation of the function using the given transformations. 12. A quadratic function with a vertical stretch of 3, shifts down 3, and translate right An inverse variation function that shifts up 3, has a vertical stretch of 5, and shifts left A radical function that shifts down 13 and right Given that x and y vary inversely and x = 10 when y = 15, find the value of the constant of variation (k) and the inverse variation equation. 16. If x and y vary directly and x = 13 when y = 20, find x when y = If x and y vary inversely and x = 13 when y = 20, find x when y = The cost to rent a ballroom for a fundraiser varies inversely with the number of people who are going to attend. If 100 people attend it will cost $20 a person. How many people would need to attend in order to get the cost down to $15 a person? 46