Exploring and Generalizing Transformations of Functions


 Daniella Cross
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1 Exploring and Generalizing Transformations of Functions In Algebra 1 and Algebra 2, you have studied transformations of functions. Today, you will revisit and generalize that knowledge. Goals: The goals of this exploration are: 1. To create a resource guide regarding different types of functions (called families of functions). 2. To gain an understanding of the similarities and differences between families of functions. 3. To generalize the transformations to any function f(x). Directions: 1) Graphing: Use a graphing calculator to do all the graphing. This will save you precious time and energy! Graph the parent function in y1. Graph the transformed function one at a time in y2. This will help you see the changes in the functions. Graph the transformed functions in order. You do not have to draw any of your graphs. However, you may wish to sketch a few for future use. Look for patterns (similarities/differences in and between sets) that will help you answer the questions.***** 2) Completing the exploration: Your results/answers to any italicized questions or directions lead to the main ideas of the exploration. Keep these results in your notes. If you find that you are repeating an answer, feel free to refer to the previous answer. The results should be in the order of the directions/questions. The results/answers should be written in complete sentences. Any tables may be completed on this sheet. 3) Submitting the exploration: You will show your understanding of this exploration by completing and submitting the Exploring Transformations of Functions Worksheet. ***** If you already know the patterns or figure out the pattern before you have graphed all the transformed functions, you may go directly to answering the questions.
2 Linear functions: Parent function: y = x Transformed function: y = ax + b (often learned as y = mx + b) Set A: Set B: Set C: Set D: y = 2x y = 2x y = x + 2 y = x  2 y = 5x y = 5x y = x + 5 y = x  5 y = 10x y =  10x y = x + 10 y = x  10 y =.5x y = .5x y =.05x 1. If a is positive will the graph be increasing or decreasing? Why? 2. If a is negative, will the graph be increasing or decreasing? Why? 3. If a >1 (If a is a large negative OR positive number), how will the graph change? 4. If a < 1 (If a is a small negative OR positive number), how will the graph change? 5. If b is positive, how is the graph transformed? 6. If b is negative, how will the graph be transformed? 7. Describe how you would recognize a linear graph. 8. Complete the tables below: y = 3x What is the value of a? y = 5x + 4 What is the value of a? y = 10x + 3 What is the value of a? X Y X Y X Y Quadratic and Cubic Functions: Part 1  Parent functions: y = x 2 and y = x 3 Transformed standard form functions: y = a x 2 + bx + c and y = ax 3 + bx 2 + cx + d Set A y = 2x 2 y = x 2 y = 2x 2 y = 10x 2 y = 0.5x 2 y = 0.05x 2 Set B y = 2x 3 y = x 3 y = 5x 3 y = 4x 3 y =.25x 3 Set C y = x 2 = 5 y = x y = x 2 10 y = x Set D y = x y = x y = x 3 6 y = x 3 6.2
3 1. If a is positive, how is the parent function transformed? 2. If a is negative, how is the parent function transformed? 3. If a > 1, how is the parent function transformed? 4. If a < 1, how is the parent function transformed? 5. If c in a quadratic function or d in a cubic function is positive, how is the graph transformed? 6. If c in a quadratic function or d in a cubic function is negative, how is the graph transformed? HONORS: 7. In a quadratic function, how do a and b interact to transform the graph? Be detailed while conscious of time! 8. Now, experiment with several cubic functions in which a, b, c and d are all changed at the same time. What do you notice? You might look carefully at the point of inflection. Part 2 Parent function: y = x 2 and y = x 3 Transformed (alternate form) function: y = a(x h) 2 + k and y = a(x h) 3 + k First, make sure you know what the h and k values actually are: Equation hvalue kvalue y = (x 19) y = (x 5) y = (x 6) 3 11 y = (x + 12) y = (x +13) Now, graph: Set A y = (x 5) 2 y = (x 6) 2 y = (x +4) 2 y = (x + 3) 2 y = (x 2) 3 y = (x 1) 3 y = (x + 7) 3 Set B y = x y = x y = x 2 4 y = x 2 7 y = x y = x 3 3 y = x y = x 3 9 Set C y = (x 5) y = (x 6) y = (x +4) 24 y = (x + 3) 27 y = (x 2) y = (x 1) 36 y = (x + 7) 31 Set D y = 3(x 5) 2 y = 3(x 6) 2 y = 2(x +4) 2 y = 2(x + 3) 2 y = 5(x 2) 3 y = 5(x 1) 3 y = 0.5(x + 7) 3 8. How do the h and k values affect the graph of y = x 2? How do the h and k values affect the graph of y = x 3?
4 9. How does the avalue affect the parent functions? How do the a and k values in these functions relate to the a and c (or d) values in part 1? 10. If we had a function y = x n, how would the function s graph be changed by y = a(x h) n + k? Square Root Function Parent function: y = Transformed function: y = a + k Set A y =3 y = 3 y = 10 y = 5 y = ½ Set B y = y = y = y = y = Set C y = + 7 y = + 3 y =  3 y = y = Set D y = y = y = 25 y = ½  8 y =  ½ How are the transformations in this section like the transformations you have seen in the earlier sections of the exploration? 2. When we graph : y = (x), we see a curve that starts at (0,0). When we make the table of : y = (x), we see that we get the result error when x < 0. Why do these results occur? 3. If the function is transformed with an h and k value other than zero, the starting point of the graph is not (0,0). Explain how you would know where the function starts. Explain why you know the function starts at this point. Absolute Value function: Parent function: y = x (on the calculator, this is y = abs(x)) Transformed function: y = a x h + k Set A y = 4 x y = 4 x y = 2 x y = 5 x y = ½ x y = 0.3 x Set B y = x 4 y = x 7 y = x + 6 y = x + 3 y = x 3.6 Set C y = x + 6 y = x + 8 y = x  3 y = x  9 y = x Set D y = 3 x y = 4 x y = 2 x y =  ½ x y = 5 x How are the transformations in this section like the transformations you have seen in the earlier sections of the exploration? 2. When we graph y = a x h + k, how can we know where the vertex will occur? How is this similar to a quadratic function graph s vertex? To a cubic function graph s point of inflection?
5 Exponential Functions An exponential function can be transformed in many ways. The standard equation y = (a)b x is often written as y = (y0)b x. Let s see how a (or y0) affects the graph of an exponential function. Part 1 a. Given each function below, identify the a (or y0 )value. Write your answers on the lines provided. y = 3(2 x ) y0 = y = 4(2 x ) a = y = 10(2 x ) y0 = y =.35(2 x ) a = b. Based on your answers to part a, what is the y0 value in y = 2 x? y0 = Graph to explore: Set A y = 3(2 x ) y = 4(2 x ) y = 10(2 x ) y =.35(2 x ) y = 3(2 x ) y = .5(2 x ) 1. How does the avalue (also called the y0 value) seem to affect these graphs? 2. How is this transformation similar and/or different than the transformations we have seen earlier in this exploration? HONORS Questions: How are y = 2 x and y = (2) x different? What changes do the ( ) make in the meaning of the expression? Try to graph y = (2) x by hand (your calculator may lock up). Why is graphing a smooth curve for this equation an impossible task?
6 Part 2 The basic transformed function is: y = (a)b (x h) + k (also written as y = (y0) b (x h) + k). Let s see how h and k affect the graphs. Experiment with these sets of function: Set A y = 2 y = 2 y = 2 y = 2 y = 2 y = 2 (x 6) (x 10) (x +1) (x + 6) (x + 10) (x 1) Set B y = 2 x + 1 y = 2 x + 5 y = 2 x + 8 y = 2 x 3 y = 2 x 7 1. How are these transformations similar those seen in the earlier sections of the exploration? Part 3 Now, let s explore the bvalue. a. Graph y = 2 x. Now graph y = 3 x in the same window so you can see both graphs at the same time. b. What point do the two graphs have in common? c. Based on your explorations in part 1, why does it makes sense that the two graphs would have this point in common? (Hint: What is the y0 value for each graph?) d. Repeat with the functions y = 5 x, y = (1/2) x and y = (1/3) x. e. What point do all these graphs have in common? Why? f. Using your results from the explorations above, fill in the blanks in the paragraph below. If the bvalue of an exponential function is greater than 1 (b > 1), then the graph is and the graph looks like (sketch a graph)
7 If the bvalue of an exponential function is less than 1 but greater than 0 (0 < b < 1), then the graph is and the graph looks like (sketch a graph) 1. How is the transformation by b different than in the other sections of this exploration? HONORS Questions: What would the graph of y = a(1) x look like? Why did you get this result? This graph represents a function. Logarithmic Functions The general function is y = loga x The parent functions we will use are: y = and y = ln(x) One transformed functions are: y = a and y =a ln(x h) + k Recall that = log x both in writing and on our calculator. Set A Set B Set C y = 5log (x) y = log (x + 2) y = log (x) + 2 y =5 ln (x) y = ln (x + 2) y = ln (x) + 2 y = 3log (x) y = log (x  4) y = log (x) 4 y = 3ln (x) y = ln (x +7) y = ln (x) + 7 y = 2log (x) y = 2 ln (x) y =  ½ log (x) y =  ½ ln (x) 1. How do a, h, and k affect the parent functions graphs? 2. How are these transformations like those we have seen previously?
8 Comparing y = f(x) and y = f(x) In the MODE menu, choose sequential in the sixth row so you can see the graphs in order. Graph y = x 3 Compare to the graph of y =  x 3 Compare to the graph of y = x3 Repeat with: y = ln(x) y = ln(x) y = ln(x) Repeat with y = (x+4) 2 y = (x+4) 2 y = (x+4) 2 1. In general, how is the graph of y = f(x) transformed when you graph y = f(x)? How is the graph of y = f(x) transformed when you graph y = f(x)?
9 Exploring Transformations of Functions Worksheet Without a calculator, match the equations with their graphs. 1. A. y = 3x + 1 B. y = 3x + 1 C. y = 3x 1 D. y = 3x 1 2. A. y = 0.5(x +3) B. y = 0.5(x 3) C. y = 0.5(x +3) 2 2 D. y = 0.5(x 3) 2 2
10 3. A. y = ln(x 7) + 1 B. y = ln(x 7) 1 C. y = ln(x 7) 1 D. y = ln(x + 7) 1 E. y = ln(x + 7) The graph below represents y = f(x). On the same axes, graph y = f(x).
11 5. The graph below represents y = f(x). On the same axes, graph y = f(x). 6. The graph below represents y = f(x). On the same axes, graph y = f(x 3)  2. (Hint: Consider Order of Operations: the 3 is subtracted from x first, then 2 is subtracted from the result.)
12 7. The graph below represents the function g(x). On the same axes, graph y = g(x +2) + 5 (Hint: Consider Order of Operations: the 2 is added to x first, then g(x+2) is multiplied by 1, then the 5 is added to the result.)
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