Module 2 Function Transformations, Pathways Algebra II Module 2 Function Transformations, Combinations, and Composition Part 2: Investigations #1-4 Investigation #1: Function Transformations: Transforming the Dependent Quantity Investigation #2: Function Transformations: Horizontal Shifts (Translations) Investigation #3: Function Transformations: Other Horizontal Transformations Investigation #4: Function Transformations: Extra Practice © 2017 Carlson & O’Bryan

Function Transformations: Transforming the Dependent Quantity Pathways Algebra II Investigation 1 Function Transformations: Transforming the Dependent Quantity © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Think about two functions f and g such that, for all x, g(x) = f (x). Then every ordered pair (x, y) in function f is also an ordered pair for g. The meaning of the statement g(x) = f (x) follows. It’s important to really think carefully about how function notation in the equation represents the relationship between the two functions. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Function h is related to function f in the following way: h(x) = 3·f (x). Explain how the ordered pairs for h are related to the ordered pairs for f. In other words, if (a, b) is an ordered pair for f, then (a, 3b) is always an ordered pair for h. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Function h is related to function f in the following way: h(x) = 3·f (x). Based on your answer to part (a), complete the following tables of values for f and h. x f (x)   h(x) –6 108 –2 84 1 –11 3 –81 –20 8 –60 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Function h is related to function f in the following way: h(x) = 3·f (x). Based on your answer to part (a), complete the following tables of values for f and h. x f (x)   h(x) –6 108 324 –2 28 84 1 –11 –33 3 –27 –81 8 –20 –60 © 2017 Carlson & O’Bryan Inv 2.1

f (x) = Pathways Algebra II Function h is related to function f in the following way: h(x) = 3·f (x). Rewrite the original equation relating the functions to show the same relationship solved for f (x) and explain what information the equation communicates. f (x) = If f (x) = x2 – 12x, what is the formula defining function h? © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions f and g are related in the following way: g(x) = ½ · f (x). Explain how the ordered pairs for g are related to the ordered pairs for f. In other words, if (a, b) is an ordered pair for f, then (a, ½b) is always an ordered pair for g. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II f (x) = | x |, and its graph is given. Draw the graph of g. c. What is the formula defining function g? © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II f (x) = | x |, and its graph is given. Draw the graph of g. c. What is the formula defining function g? © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II In Exercises #1-2 we explored examples of function transformations. A function transformation is a way to define a new function based on the behavior of some known function. Specifically, the transformations in Exercises #1-2 were a vertical stretch and vertical compression respectively. The “vertical” name is used because the only differences between the two functions is in the values represented on the vertical axis for each input value, and “stretch” or “compression” is used because the output values are scaled by some factor so that the function appears to stretch away from or compress towards the horizontal axis when the function are graphed. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II There are many other types of transformations because there are many ways that a new function can be created by modifying a known function. Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. Explain how the ordered pairs for p are related to the ordered pairs for h. In other words, if (a, b) is an ordered pair for h, then (a, b + 4) is always an ordered pair for p. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. Use your answer to part (a) to complete the following tables of values. x h(x)   p(x) 4 1 2 1.414 6 2.828 8 6.828 9 3 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. Use your answer to part (a) to complete the following tables of values. x h(x)   p(x) 4 1 5 2 1.414 5.414 6 8 2.828 6.828 9 3 7 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. Explain how the ordered pairs for r are related to the ordered pairs for h. In other words, if (a, b) is an ordered pair for h, then (a, b – 2) is always an ordered pair for r. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. The graph of h is given. Use your answer to part (c) to draw a graph for r. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. The graph of h is given. Use your answer to part (c) to draw a graph for r. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Functions h, p, and r are related in the following ways: p(x) = h(x) + 4 and r(x) = h(x) – 2. Given that f (x) = √x, write the formulas defining functions p and r. The transformations in Exercise #3 are called vertical shifts or vertical translations. Two functions related by a vertical shift have a constant difference in their output values for every given input value. Let’s examine one more type of vertical transformation for defining a new function from a known function. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions g and h are related in the following way: h(x) = – g(x). Explain how the ordered pairs for h are related to the ordered pairs for g. In other words, if (a, b) is an ordered pair for g, then (a, –b) is always an ordered pair for h. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions g and h are related in the following way: h(x) = – g(x). Use your answer to part (a) to complete the following tables of values. x g(x)   h(x) –3.8 –9.7 –1 –5.5 2.5 6 5 8 12.4 14.6 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions g and h are related in the following way: h(x) = – g(x). Use your answer to part (a) to complete the following tables of values. x g(x)   h(x) –3.8 –9.7 9.7 –1 –5.5 5.5 2.5 6 5 –5 8 –8 12.4 14.6 –14.6 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions g and h are related in the following way: h(x) = – g(x). The graph of g is given. Use your answer to part (c) to draw a graph for h. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions g and h are related in the following way: h(x) = – g(x). The graph of g is given. Use your answer to part (c) to draw a graph for h. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Two functions g and h are related in the following way: h(x) = – g(x). What is the formula for g? What is the formula for h? © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II The transformation in Exercise #4 is called a vertical reflection over the horizontal axis. It is possible to reflect vertically over other horizontal lines, although we will not discuss those types of transformations at this time. Two functions related by a vertical reflection over the horizontal axis have output values with the same magnitude for all input values, but their signs are opposite. © 2017 Carlson & O’Bryan Inv 2.1

Transformations of the Dependent Quantity Pathways Algebra II Transformations of the Dependent Quantity Two functions f and g are related by a vertical shift (or vertical translation) if, for all x, the output values of the functions have a constant difference (the output value of one function is the same amount more or less than the output of the other function for all x). © 2017 Carlson & O’Bryan Inv 2.1

Transformations of the Dependent Quantity Pathways Algebra II Transformations of the Dependent Quantity Two functions f and g are related by a vertical stretch/compression if, for all x, the output values of the functions have a constant ratio (the output value of one function is the same number of times as large as the output of the other function for all x). © 2017 Carlson & O’Bryan Inv 2.1

Transformations of the Dependent Quantity Pathways Algebra II Transformations of the Dependent Quantity Two functions related by a vertical reflection across the horizontal axis if, for all x, the output values of the two functions have the same magnitude (absolute value) but opposite signs. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II New functions can also be created by combining multiple transformations. When thinking about how the functions’ ordered pairs are related, think about evaluating the formulas according to the order of operations. For example, let f (x) = x2 + 3 and let’s define a new function g such that g(x) = – f (x) + 5. Then: © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II For example, let f (x) = x2 + 3 and let’s define a new function g such that g(x) = – f (x) + 5. Then: If we want to know the values of g(–4) and g(6), we have to think about how these values are determined using the “known” function. g(–4) = – f (–4) + 5 g(6) = – f (6) + 5 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II For example, let f (x) = x2 + 3 and let’s define a new function g such that g(x) = – f (x) + 5. 5. a. Determine the values of f (–4) and f (6). Determine the values of g(–4) and g(6) based on the given information. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II For example, let f (x) = x2 + 3 and let’s define a new function g such that g(x) = – f (x) + 5. The graph of f is given. Draw the graph of g. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II For example, let f (x) = x2 + 3 and let’s define a new function g such that g(x) = – f (x) + 5. The graph of f is given. Draw the graph of g. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II For example, let f (x) = x2 + 3 and let’s define a new function g such that g(x) = – f (x) + 5. d. Write the formula for g. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Given a table of values for f, create a table of values for g and h if g(x) = –2.5 · f (x) + 1 and h(x) = 1/3 · f (x) – 4. x f (x)   g(x) h(x) –6.5 15 –4.2 6 –1 –4 0.2 –21 2.6 –8 9.8 1 © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Given a table of values for f, create a table of values for g and h if g(x) = –2.5 · f (x) + 1 and h(x) = 1/3 · f (x) – 4. x f (x)   g(x) h(x) –6.5 15 –36.5 1 –4.2 6 –14 –2 –1 –4 11 –16/3 0.2 –21 53.5 –11 2.6 –8 21 –20/3 9.8 –1.5 –11/3 The output of g at x is 1 more than –2.5 times as large as the output of f at the same value of x. The output of h at x is 4 less than 1/3 times as large as the output of f at the same value of x. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Given the graph of f, draw the graphs of g and h if g(x) = – f (x) – 2 and h(x) = 1.5 · f (x) + 2. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Given the graph of f, draw the graphs of g and h if g(x) = – f (x) – 2 and h(x) = 1.5 · f (x) + 2. © 2017 Carlson & O’Bryan Inv 2.1

Pathways Algebra II Given the graphs of f, g, and h, complete the following. Use function notation to express g in terms of f. Use function notation to express h in terms of f. g(x) = h(x) = © 2017 Carlson & O’Bryan Inv 2.1

Function Transformations: Horizontal Shifts (Translations) Pathways Algebra II Investigation 2 Function Transformations: Horizontal Shifts (Translations) © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Let’s compare the functions f (x) = 2x – 5 and g(x) = 2(x – 3) – 5. They have the same rate of change but they have different given reference points. The two functions’ behavior is identical except the behavior occurs with reference to x = 0 in f and with reference to x = 3 in g. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Use the graphs or formulas to determine which of the following statements are true and which are false. a. g(5) = f (2) b. g(7) = f (4) c. g(1) = f (4) d. g(4) = f (1) e. g(2) = f (–1) f. g(x) = f (x + 3)   g. g(x) = f (x – 3) h. f (x) = g(x + 3) i. f (x) = g(x – 3) true true false true true false true true false © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 2. The graph of f is given. Let g(x) = f (x – 6). a. Use the graph to evaluate g(6). b. Use the graph to evaluate g(8). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 2. The graph of f is given. Let g(x) = f (x – 6). c. Use the graph to evaluate g(4.5). d. Draw the graph for g. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 2. The graph of f is given. Let g(x) = f (x – 6). c. Use the graph to evaluate g(4.5). d. Draw the graph for g. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). x f (x)   g(x) h(x) –7 28 –4.5 20.5 –2 13 7 2 1 3.25 –2.75 Complete each of the following statements (not all of these may relate to the table of values). g(6) = f ( ) g(–9) = f ( ) g(0) = f ( ) g(3.25) = f ( ) © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). x f (x)   g(x) h(x) –7 28 –4.5 20.5 –2 13 7 2 1 3.25 –2.75 Complete each of the following statements (not all of these may relate to the table of values). g(6) = f (8) g(–9) = f (–7) g(0) = f (2) g(3.25) = f (5.25) © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). x f (x)   g(x) h(x) –7 28 –4.5 20.5 –2 13 7 2 1 3.25 –2.75 Fill in a table of values for g. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). x f (x)   g(x) h(x) –7 28 –9 –4.5 20.5 –6.5 –2 13 –4 7 2 1 3.25 –2.75 1.25 Fill in a table of values for g. The outputs of g and f will be the same when the argument x to g is 3 units less than the argument x + 2 to f (or the argument to f is 2 units more than the argument to g). Thus, to create the table for g we keep the same set of outputs but decrease the corresponding inputs by 2. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). x f (x)   g(x) h(x) –7 28 –9 –4.5 20.5 –6.5 –2 13 –4 7 2 1 3.25 –2.75 1.25 If h(x) = f (x + 5), fill in the table of values for h. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). x f (x)   g(x) h(x) –7 28 –9 –12 –4.5 20.5 –6.5 –9.5 –2 13 –4 7 –5 2 1 –3 3.25 –2.75 1.25 –1.75 If h(x) = f (x + 5), fill in the table of values for h. The outputs of h and f will be the same when the argument x to h is 5 units less than the argument x + 5 to f (or the argument to f is 5 units more than the argument to h). Thus, to create the table for h we keep the same set of outputs but decrease the corresponding inputs by 5. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 3. A table of values for f is given. Let g(x) = f (x + 2). If f (x) = –3x + 7, write the formulas for g and h. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II The transformations in Exercises #1-3 are examples of horizontal shifts or horizontal translations because new functions use the same pattern of outputs but for different input values. Specifically, the input values differ by a constant amount to produce the same output value. © 2017 Carlson & O’Bryan Inv 2.2

Transformations of the Independent Quantity: Horizontal Translations Pathways Algebra II Transformations of the Independent Quantity: Horizontal Translations Two functions f and g are related by a horizontal shift (or horizontal translation) if the same pattern of output values define each function but for inputs with a constant difference. When k > 0, the graph of g appears to be the graph of f shifted to the right k units. This is because g needs larger input values to produce the same output value. When k < 0, the graph of g appears to be the graph of f shifted to the left k units. This is because g needs smaller input values to produce the same output value. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II In applied problems we see horizontal shifts most commonly when we want to change the definition of our input quantity so that we think about the reference point differently. We’ll explore this idea in the next set of Exercises. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II The following graph models the student enrollment at a certain high school as a function of the number of years since 1990. [Call this function f.] © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Draw a new graph representing function g, the student enrollment at the high school as a function of the number of years since 1980. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Draw a new graph representing function g, the student enrollment at the high school as a function of the number of years since 1980. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Which of the following statements is true? Defend your choice. g(1) = f (11) or g(1) = f (–9) g(1) = f (–9); The expected enrollment 1 year after 1980 (so in 1981) is the same as the expected enrollment 9 years before 1990 (so in 1981). What is the relationship between the graphs of f and g? The graph of g is the graph of f but shifted to the right 10 units because we need larger inputs to g to produce the desired enrollment information. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II If x represents the number of years since 1980, which of the following statements is true? Defend your choice. g(x) = f (x + 10) or g(x) = f (x – 10) Suppose we want to find the student enrollment in the year 1995. Then we would evaluate g(15). Using function f, however, we would need to evaluate f (5) since f tells us the expected enrollment based on some number of years since 1990 instead of 1980. So g(15) = f (5) = f (15 – 10). Or suppose we want to find the enrollment in 1980. Then we would evaluate g(0). Using function f, however, we would need to evaluate f (–10). So g(0) = f (–10) = f (0 – 10), and in general g(x) = f (x – 10). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Draw a new graph representing function h, the student enrollment at the high school as a function of the number of years since 1995. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Draw a new graph representing function h, the student enrollment at the high school as a function of the number of years since 1995. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Which of the following statements is true? Defend your choice. h(7) = f (12) or h(7) = f (2) The expected enrollment 7 years after 1995 (so in 2002) is the same as the expected enrollment 12 years after 1990 (so in 2002). What is the relationship between the graphs of f and h? The graph of h is the graph of f but shifted to the left 5 units because we need smaller inputs to yield the desired enrollment information. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II If t represents the number of years since 1995, which of the following statements is true? Defend your choice. h(t) = f (t + 5) or h(t) = f (t – 5) Suppose we want to find the student enrollment in the year 1995. Then we would evaluate h(0). Using function f, however, we would need to evaluate f (5) since f tells us the expected enrollment based on some number of years since 1990 instead of 1995. So h(0) = f (5) = f (0 + 5). Or suppose we want to find the enrollment in 1980. Then we would evaluate h(–15). Using function f, however, we would need to evaluate f (–10). So h(–15) = f (–10) = f (–15 + 5), and in general h(t) = f (t + 5). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 5. The graph of f is given. a. Draw the graph of g given that g(x) = f (x – 2). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 5. The graph of f is given. a. Draw the graph of g given that g(x) = f (x – 2). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 5. The graph of f is given. Draw the graph of h given that h(x) = f (x + 1) + 2. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 5. The graph of f is given. Draw the graph of h given that h(x) = f (x + 1) + 2. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 5. The graph of f is given. Draw the graph of j given that j(x) = – f (x + 4). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II 5. The graph of f is given. Draw the graph of j given that j(x) = – f (x + 4). © 2017 Carlson & O’Bryan Inv 2.2

b. Create a table of values for h if h(x) = –2 f (x – 4). Pathways Algebra II The table below shows ordered pairs for function f. Create a table of values for g if g(x) = f (x + 3) – 2. x f (x)   g(x) h(x) –2 16 –1 14 8 1 4 2 3 26 © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II The table below shows ordered pairs for function f. Create a table of values for g if g(x) = f (x + 3) – 2. x f (x)   g(x) h(x) –2 16 2 14 –1 3 12 8 4 6 1 5 26 7 24 The outputs of g and f will be differ by 2 when the argument x to g is 3 units less than the argument x + 3 to f (or the argument to f is 3 units more than the argument to g). We can think about creating the table in two steps. First, we keep the same set of outputs but decrease the corresponding inputs by 3. Then we decrease the output values for each input value by 2. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II The table below shows ordered pairs for function f. Create a table of values for h if h(x) = –2 f (x – 4). x f (x)   g(x) h(x) –2 16 2 14 –1 3 12 8 4 6 1 5 26 7 24 © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II The table below shows ordered pairs for function f. Create a table of values for h if h(x) = –2 f (x – 4). x f (x)   g(x) h(x) –2 16 2 14 –6 –32 –1 3 12 –5 –28 8 4 6 –4 –16 1 5 –3 –8 26 7 24 –52 The output of h will be –2 times as large as the output of f when the argument x to h is 4 units more than the argument x – 4 to f (or the argument to f is 4 units less than the argument to h). We can think about creating the table in two steps. First, we keep the same set of outputs but increase the corresponding inputs by 4. Then we multiply all of the the output values for each input value by –2. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Given that f (x) = and g(x) = –2x2 + 3, complete the following. Evaluate h(2) if h(x) = f (x + 5). Evaluate j(–1) if j(x) = g(x – 3). © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Given that f (x) = and g(x) = –2x2 + 3, complete the following. Write the formula defining p if p(x) = 3 f (x – 1). Write the formula defining q if q(x) = g(x + 4) + 2. © 2017 Carlson & O’Bryan Inv 2.2

Pathways Algebra II Use the given graphs to complete the following. Use function notation to represent g in terms of f. Use function notation to represent h in terms of f. g(x) = h(x) = © 2017 Carlson & O’Bryan Inv 2.2

Function Transformations: Other Horizontal Transformations Pathways Algebra II Investigation 3 Function Transformations: Other Horizontal Transformations © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Imagine filling a certain vase from a faucet pouring out water at a constant rate. The following graph represents the height of water in the vase h, in inches, as a function of the time elapsed t, in seconds, since the vase began filling. Call this function f with h = f (t). © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Using the graph, evaluate each of the following or solve the equation and explain the meaning of your answers. Answers may vary based on student estimates. a. f (5) b. f (8) 3.3; 5 seconds into filling the vase the height of water is 3.3 inches 4.6; 8 seconds into filling the vase the height of water is 4.6 inches c. f (t) = 3 d. f (t) = 6 t = 4; 4 seconds into filling the vase the height of water is 3 inches t = 9.8; 9.8 seconds into filling the vase the height of water is 6 inches © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II After letting the vase fill to capacity, we empty it and change the faucet so that the water is pouring out at half the previous rate. Discuss with a partner how this changes the situation. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II For the new rate described in Exercise #2, let h = g(t) model the relationship between the height of water in the vase in inches and the time elapsed since the vase began filling. How long will it take for the height of water to reach 3 inches? How do you know? 8 seconds; Originally it took 4 seconds for the water height to reach 3 inches. Since the water is flowing in at half the previous rate then it takes twice as long to reach that height. How long will it take for the height of water to reach 6 inches? about 19.6 seconds; Originally it took 9.8 seconds for the water height to reach 6 inches. Since the water is flowing in at half the previous rate then it takes twice as long to reach that height. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II c. Complete each of the following statements. i. g( ) = f (2) ii. g( ) = f (3.25) iii. g(11) = f ( ) iv. g(x) = f ( ) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II c. Complete each of the following statements. i. g(4) = f (2) ii. g(6.5) = f (3.25) iii. g(11) = f (5.5) iv. g(x) = f (1/2 x) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Draw the graph of g on the same axes with f and describe their relationship. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Draw the graph of g on the same axes with f and describe their relationship. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II a. If instead the faucet’s rate is tripled from the original situation (call the new function j so h = j(t) for this rate of flow), how long will it take for the height of water to reach 3 inches? How do you know? 4/3 seconds; Originally it took 4 seconds for the water height to reach 3 inches. Since the water is flowing at three times the original rate then it takes 1/3 times as much time to reach that height. How long will it take for the height of water to reach 6 inches? 9.8/3 (or about 3.3) seconds; Originally it took 9.8 seconds for the water height to reach 3 inches. Since the water is flowing at three times the original rate then it takes 1/3 times as much time to reach that height. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Complete the following statements. i. j( ) = f (2) ii. j( ) = f (9) iii. j(1.5) = f ( ) iv. j(x) = f ( ) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Complete the following statements. i. j(2/3) = f (2) ii. j(3) = f (9) iii. j(1.5) = f (4.5) iv. j(x) = f (3x) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Draw the graph of j on the same axes as f and describe their relationship. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Draw the graph of j on the same axes as f and describe their relationship. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II The graphs of f and g and f and h are given. Use these graphs to answer the questions that follow. a. g( ) = f (–2) b. g( ) = f (–1) c. g(–18) = f ( ) d. g(x) = f ( ) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II The graphs of f and g and f and h are given. Use these graphs to answer the questions that follow. a. g(–6) = f (–2) b. g(–3) = f (–1) c. g(–18) = f (–6) d. g(x) = f (1/3 x) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II The graphs of f and g and f and h are given. Use these graphs to answer the questions that follow. e. h( ) = f (–1.5) f. h( ) = f (–2) g. h(–4) = f ( ) h. h(x) = f ( ) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II The graphs of f and g and f and h are given. Use these graphs to answer the questions that follow. e. h(–1) = f (–1.5) f. h(–4/3) = f (–2) g. h(–4) = f (–6) h. h(x) = f (1.5x) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Transformations of the Independent Quantity: Horizontal Stretches/Compressions Two functions f and g are related by a horizontal stretch/compression if the same pattern of output values define each function but for inputs related by a constant ratio. When k > 1, the graph of g appears to be the graph of f compressed towards the vertical axis by a factor of 1/k. This is because g needs input values 1/k times as large to produce the same output value. When k < 1, the graph of g appears to be the graph of f stretched away from the vertical axis by a factor of k. This is because g needs input values k times as large to produce the same output value. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Transformations of the Independent Quantity: Horizontal Stretches/Compressions Ex. 1: If g(x) = f (4x), then the graph of g will appear to be the graph of f compressed towards the vertical axis by a factor of ¼. This is because g needs input values ¼ times as large as the input to f to produce the same output value. Ex. 2: If g(x) = f (1/3 x), then the graph of g will appear to be the graph of f stretched away from the vertical axis by a factor of 3. This is because g needs input values 3 times as large as the input to f to produce the same output value. © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II It is also possible to construct new functions through a horizontal reflection across the vertical axis. © 2017 Carlson & O’Bryan Inv 2.3

Transformations of the Independent Quantity: Horizontal Reflections Pathways Algebra II Transformations of the Independent Quantity: Horizontal Reflections Two functions f and g are related by a horizontal reflection across the vertical axis if the same output values define each function but for inputs with the opposite sign. If g(x) = f (–x), then each of the following statements is true. g(2) = f (–2) g(4) = f (–4) g(–1) = f (1) g(–3) = f (3) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II 6. For each given function f, create the graph of the new function. a. g(x) = 3 f (–x) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II 6. For each given function f, create the graph of the new function. a. g(x) = 3 f (–x) © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II 6. For each given function f, create the graph of the new function. b. g(x) = – f (4x) + 1 © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II 6. For each given function f, create the graph of the new function. b. g(x) = – f (4x) + 1 © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –12 24 –6 8 –3 15 –2 36 4 a. g(x) = f (–2x) b. c. j(x) = 2 f (x + 4) – 1 © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –12 24 –18 4 –6 8 –7.5 –2 –3 15 3 36 6 a. g(x) = f (–2x) b. c. j(x) = 2 f (x + 4) – 1 © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –12 24 –18 4 –36 –6 8 –7.5 –2 –4 –3 1.5 15 3 45 1 36 6 108 a. g(x) = f (–2x) b. c. j(x) = 2 f (x + 4) – 1 © 2017 Carlson & O’Bryan Inv 2.3

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –12 24 –18 4 –36 –16 47 –6 8 –7.5 –2 –4 –10 15 –3 1.5 –7 3 45 1 11 –5 36 6 108 32 7 a. g(x) = f (–2x) b. c. j(x) = 2 f (x + 4) – 1 © 2017 Carlson & O’Bryan Inv 2.3

Function Transformations: Pathways Algebra II Investigation 4 Function Transformations: Extra Practice © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Use each given function to graph the new transformed function. a. g(x) = f (–x) + 3 b. g(x) = ½ f (3x) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Use each given function to graph the new transformed function. a. g(x) = f (–x) + 3 b. g(x) = ½ f (3x) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Use each given function to graph the new transformed function. a. g(x) = f (–x) + 3 b. g(x) = ½ f (3x) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II c. g(x) = – f (x + 4) d. g(x) = f (½ x) – 2 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II c. g(x) = – f (x + 4) d. g(x) = f (½ x) – 2 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II c. g(x) = – f (x + 4) d. g(x) = f (½ x) – 2 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II For each of the following, describe the transformation(s) on f needed to generate function g. g can be generated by horizontally compressing f by a factor of 1/3, vertically stretching the result by a factor of 4, and then vertically reflecting the result across the horizontal axis. g can be generated by horizontally shifting f to the right 7 units, horizontally reflecting the result across the vertical axis, and shifting the result up 8 units. © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II For each of the following, describe the transformation(s) on f needed to generate function g. g can be generated by horizontally stretching f by a factor of 3/2, or 1.5, and then vertically compressing the result by a factor of ½. g can be generated by horizontally shifting f to the left 1 unit, vertically stretching the result by a factor of 3, and then shifting the result down 5 units. © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II For each of the following, describe the transformation(s) on f needed to generate function g. g can be generated by horizontally shifting f to the right 2 units and vertically shifting the result up 4 units. g can be generated by horizontally shifting f to the right 3 units and then vertically stretching the result by a factor of 2. © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –8 –1 –2 3 7 1 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –8 –1 –16 –7 –2 3 –4 –3 7 1 0.5 –6 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –8 –1 –16 –7 –4 3 –2 –3 2 7 1 4 –5 0.5 –6 5 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Given the table of values for f, create a table of values for each other function. x f (x)   g(x) h(x) j(x) –8 –1 –16 –7 –4 3 –0.25 –2 –3 2 7 1 4 –5 0.5 –6 5 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Given that f (x) = | x |, g(x) = 2(x – 3) + 1, and h(x) = (x + 1)2, write the formula for each of the following functions. a. p if p(x) = 3 f (x + 4) – 8 b. n if n(x) = g(0.5x) – 7 © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Given that f (x) = | x |, g(x) = 2(x – 3) + 1, and h(x) = (x + 1)2, write the formula for each of the following functions. d if d can be generated by vertically reflecting h over the horizontal axis after shifting it to the right 5 units. r if r can be generated by horizontally compressing f by a factor of 4/7 and shifting it up 3 units. © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f models the distance between you and a bug (in feet) as it flies around the room based on the number of seconds since you first noticed it. Function g models the distance between you and the bug (in inches) based on the number of seconds since you first noticed it, t. Then: g(t) = f ( ) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f models the distance between you and a bug (in feet) as it flies around the room based on the number of seconds since you first noticed it. Function g models the distance between you and the bug (in inches) based on the number of seconds since you first noticed it, t. Then: g(t) = 12 ∙ f (t) Since inches are a smaller unit of measure than feet, the outputs of g must be 12 times as large as the outputs of f to represent the same distance. © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f models the distance between you and a bug (in feet) as it flies around the room based on the number of seconds since you first noticed it. Function h models the distance between you and the bug (in feet) based on the number of minutes since you first noticed it, x. Then: h(x) = f ( ) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f models the distance between you and a bug (in feet) as it flies around the room based on the number of seconds since you first noticed it. Function h models the distance between you and the bug (in feet) based on the number of minutes since you first noticed it, x. Then: h(x) = f (60x) Since minutes are a larger unit of measure than seconds, h needs an input 1/60 times as large as the input to f to achieve the same output (or the input to f must be 60 times as large). So, for example, h(1) = ­f (60) because the bug’s distance from you after 1 minute, represented by h(1), is the same as the bug’s distance from you after 60 seconds, represented by f (60). © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f models the distance between you and a bug (in feet) as it flies around the room based on the number of seconds since you first noticed it. Your friend noticed the bug 5 seconds before you did. Function j models the distance between you and the bug (in yards) based on the number of seconds since your friend first noticed it, n. Then: j(n) = f ( ) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f models the distance between you and a bug (in feet) as it flies around the room based on the number of seconds since you first noticed it. Your friend noticed the bug 5 seconds before you did. Function j models the distance between you and the bug (in yards) based on the number of seconds since your friend first noticed it, n. Then: j(n) = 1/3 ∙ f (n – 5) © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f with p = f (t) models p, the population of Phoenix, AZ, in terms of the number of years since the beginning of 2000, t. Describe the transformation(s) of f needed to create a new function g that models that population of Phoenix, AZ in millions of people in terms of the number of years since the beginning of 2000, t. g can be generated by vertically compressing f by a factor of Describe the transformation(s) of f needed to create a new function h that models that population of Phoenix, AZ in terms of the year, y. g can be generated by horizontally shifting f to the right 2000 units. © 2017 Carlson & O’Bryan Inv 2.4

Pathways Algebra II Function f with p = f (t) models p, the population of Phoenix, AZ, in terms of the number of years since the beginning of 2000, t. Describe the transformation(s) of f needed to create a new function j that models that population of Phoenix, AZ in millions of people in terms of the number of decades since the beginning of 2000, d. j can be generated by horizontally compressing f by a factor of 1/10 and vertically compressing the result by a factor of © 2017 Carlson & O’Bryan Inv 2.4