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1 `Name: Period: Unit 4 Modeling with Advanced Functions 1

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3 Piecewise Functions Example 1: f x, if x) x 3, if ( 2 x x 1 1 For all x s < 1, use the top graph. For all x s 1, use the bottom graph Example 2: x 2,if x 2 f(x) 2x 1,if x 2 Evaluate f(x) when x=0, x=2, x=4 First, you need to figure out which equation to use. You NEVER use both. f(0) f(2) f(4) Example 3: x 1, if x 2 f( x) x 1, if x 2 Point of Discontinuity x=2 We must use open and closed circles so the graph will pass the vertical line test. 3

4 Piecewise Functions Independent Practice 2 x 1, x 3 1) f(x) 3 2 x 3, x 3 2) 4, 3 f(x) x 2, 4 2 x 7, x 8 8 x 4 x 4 3) 3, f(x) x 2, x 4 x 4 4) 3x 4, f(x) 2x 2, x 2 x 2 4

5 Piecewise Functions Practice Homework 1. Consider the function x + 4 if x 2 f(x) = { 2 if 2 < x 3 } 4 x if x > 3 f(-1)= f(-3)= f(3.5)= Domain: Range: Graph 2. In a certain county, income tax is assessed according to the following function where T is the income tax and x is the amount of income. 0 if 0 x < 10,000 T(x) = { 0.08x if 10,000 x 20,000} x if x > 20,000 Find T(5,000), T(10,000), T(12,000) and T(25,000). What do your answers in part (a) represent? What is the income for someone paying $5350, $1300, $800? What is the domain of this function? 3. A private museum charges $40 for a group of 10 or fewer people. A group of more than 10 people must, in addition to the $40, pay $2 per person for the number of people above 10. The maximum group size is 50. What is the cost for 8 people? What is the cost for 10 people? What is the cost for 25 people? If the cost was $120, how many people were in the group? Where are the discontinuities? Write the piecewise function for the museum. 5

6 Absolute Value The function f(x) = x is an absolute value function. Absolute Value is so it is ALWAYS. The graph of this piecewise function consists of 2 rays, is v-shaped, and opens up. The highest or lowest point on the graph of an absolute value function is called the. The of the graph of a function is a vertical line that divides the graph into mirror images. The of a function f(x) are the values of x that make the value of f(x) equal to 0. General form of the Absolute Value Function 6

7 Examples Identify the transformations. y = 3 x y = x y = 2 x y = -1/3 x Graph y = -2 x Graph y = -1/2 x 1-2 Compare to y = x Compare to y = x 7. Write the function for 8. Write the function for 7

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9 Greatest Integer Function The output of the Greatest Integer Function is the integer that is the input. (Think Price is Right! Closest integer without going over.) For example, 5.3 = 5 and 4.9 = 5. Examples: = = 3..3 = = Complete the table to determine some points of the parent greatest integer function, y = x. Use them to help you graph on the grid provided. Think about when the value of the function moves from one step to the next as you graph. x y The graph of this function is much different from the other functions we have studied so far. You can see why it is often called a step function. Make sure that the endpoints of each step are graphed appropriately. If the value of the endpoint is included in the step, then it should be graphed as a closed circle. If the value of the endpoint is not included in the step, then it should be graphed as an open circle. 9

10 Transformations of the Greatest Integer Function General Form So is a starting point for your steps. The space between your steps (vertically) is. If is the steps go and if is then the steps. Example: Graph f(x) = 2 x Start Step height Example 2: Graph y = 2x Get in graphing form! Start Step height 10

11 Graph the following functions. 1. y = x 3 2. y = x 3 3. y = x y = x 5. y = x y = x

12 Real World Examples of Step Functions Prior to September, 2000, taxi fares from Washington DC to Maryland were described as follows: $2.00 up to and including ½ mile, $0.70 for each additional ½ mile increment. (10) Describe the independent and dependent variables and explain your choices. (11) Graph the fares for the first 2 miles: (Make sure to label the axes.) (12) Write the piecewise function for 0 to 2 miles. f x (13) Discuss why this is a step function and it is different from the greatest integer parent function f(x) = x. 12

13 Homework Step Function a 6. You are selling candy bars. The taxable amount and tax imposed up to $1 are shown below. For amounts between $0.01 and $0.20, the tax is $0.01 For amount greater than $0.20 and less than or equal to $0.40, the tax is $0.02 For amount greater than $0.40 and less than or equal to $0.60, the tax is $0.03 For amount greater than $0.60 and less than or equal to $0.80, the tax is $0.04 For amount greater than $0.80 and less than or equal to $1.00, the tax is $0.05 a) A candy bar costs $0.55. What is the total cost with tax? b) Your aunt purchased 3 candy bars at $0.65 each? What is the total cost? c) Someone wanted to purchase 4 candy bars for $0.75 each. If they had 3 dollars and a quarter, could they buy the candy including the tax? Explain your answer. 13

14 14 Graphing Square Root and Cube Root Make a table for each function f(x) = x 2 f(x) = x f(x) = x 3 f(x) = x 3 Ignore the points with decimals. What do you notice about the other points? These functions are of each other. By definition, this means the and the. Plot the points from the tables above. This causes the graphs to have the but to be over the line. x y x y x y x y

15 The Square Root Function Reflect the function f(x) = x 2 over the line y = x. Problems? We have to define the Square Root as. This means that we will only use the side of the graph. The result: f(x) = x Characteristics of the graph Vertex End Behavior Domain Range Symmetry Pattern General form of the Square Root Function 15

16 The Cube Root Function Reflect the function f(x) = x 3 over the line y = x. Problems? 3 The result: f(x) = x Characteristics of the graph Vertex End Behavior Domain Range Symmetry Pattern General form of the Cube Root Function 16

17 Examples: f(x) = x 3 3 f(x) = x f(x) = x 3 f(x) = 2 x + 3 Sometimes the functions are not in graphing form. We may have to use some of our algebra skills to transform the equations into something we can use. 3 f(x) = 4x 12 f(x) = 8x

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19 Inverse Variation (Rational Functions) Many people take long car trips for business, vacations, and sometimes just commuting to work. While driving at slower speeds can save gas, driving at faster speeds can save time. For example, a 300-mile trip takes 6 hours at 50 mph, but only 5 hours at 60 mph. Think about how driving time would change if an average speed of 50 mph decreased to 40 mph. Suppose that your family is planning a 250 mile trip by car to visit relatives. Your average speed could vary from as little as 20 mph to 60 mph or more. Your average speed depends on what roads you take, traffic, weather, speed limits, and the driver s preferred pace. 1. How long will that 250 mile trip take if you average: o 20 mph o 40 mph o 60 mph 2. Write a rule that gives time of this trip t as a function of the average driving speed s. a. Use your calculator to make a table showing data (speed, time) for the 250 mile trip. Then sketch the graph of this relation below. Speed (mph) Time (hours) Describe as accurately as possible, the pattern relating average speed and time of your trip. How is that pattern shown in the table and graph? 19

20 Our road trip scenario is an example of inverse variation. Inverse variation is the opposite of direct variation in that values of the two variables change in an opposite manner as one increases, the other decreases. Notice the shape of the graph of inverse variation. If the value of x, then y. If x, the y value. We say that y varies inversely as the value of x. INDEPENDENT PRACTICE 1. R varies inversely with variable T. If R is 168 when T = 24, find R when T = The volume, V, of a gas varies inversely as the pressure, p, in a container. If the volume of a gas is 200cc when the pressure is 1.6 liters per square centimeter, find the volume (to the nearest tenth) when the pressure is 2.8 liters per sq centimeter. 3. In science, one theory of life expectancy states that the lifespan of mammals varies inversely to the number of heartbeats per minute of the animal. If a gerbil's heart beats 360 times per minute and lives an average of 3.5 years, what would be the life expectancy of a human with an average of 72 beats per minute? Does this theory appear to hold for humans? 4. The values (9.7, 8) and (3, y) are from an inverse variation. Find the missing value and round to the nearest hundredth. 5. A drama club is planning a bus trip to New York City to see a Broadway play. The cost per person for the bus rental varies inversely as the number of people going on the trip. It will cost $30 per person if 44 people go on the trip. How much will it cost per person if 60 people go on the trip? Round your answer to the nearest cent, if necessary. 20

21 Joint and Combined Variation 1. If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find a when b = 9 and c = Wind resistance varies jointly as an object s surface area and velocity. If an object traveling at 40 mile per hour with a surface area of 25 square feet experiences a wind resistance of 225 Newtons, how fast must a car with 40 square feet of surface area travel in order to experience a wind resistance of 270 Newtons? 4. If y varies directly as x and inversely as z, and y = 22 when x = 4 and z = 6, find y when x = 10 and z = If p varies directly as the square of q and inversely as the square root of r, and p = 60 when q = 6 and r = 81, find p when q = 8 and r = The centrifugal force of an object moving in a circle varies jointly with the radius of the circular path and the mass of the object and inversely as the square of the time it takes to move about one full circle. A 6 gram object moving in a circle with a radius of 75 centimeters at a rate of 1 revolution every 3 seconds has a centrifugal force of 5000 dynes. Find the centrifugal force of a 14 gram object moving in a circle with radius 125 centimeters at a rate of 1 revolution every 2 seconds. 21

22 Solve Rational Equations Vocabulary: To solve a rational equation when each side is a single rational expression (a proportion), you can use. Extraneous solution: An apparent solution that must be rejected because it does not satisfy the original equation (ie. One or more of the denominators becomes 0). Solve and check answer(s): x x 3 x 4x 2. x 5 x 2 Solve rational equations that are not proportions, and check answer(s): You can solve these easily by multiplying each side of the equation by the of each rational expression x 6 3x 4 2x x x 2 6 8x 4x 2 5. x 3 x 9 x x 3x x 3 x 22

23 Solving Rational Equations HW (Kuta) 23

24 Graphing Rational Functions General form of the Rational Function A Asymptotes: x =, y = 1. 4 y Steps to Graph Rational Functions x 1 st : Draw the asymptotes x = and y =. 2 nd : Plot points to L and R of vertical asymptote (at least two on each side) 3 rd : Draw the branches 4 th : State the domain and range. Domain Range 1. 3 y Steps to Graph Rational Functions x 1 st : Draw the asymptotes x = and y =. 2 nd : Plot points to L and R of vertical asymptote (at least two on each side) 3 rd : Draw the branches 4 th : State the domain and range. Domain Range 24

25 6 3. y 2 Steps to Graph Rational Functions x 3 1 st : Draw the asymptotes x = and y =. 2 nd : Plot points to L and R of vertical asymptote 3 rd : Draw the branches 4 th : State the domain and range. Domain Range 3 4. y 1 x 2 1 st : Draw the asymptotes x = and y =. 2 nd : Plot points to L and R of vertical asymptote 3 rd : Draw the branches 4 th : State the domain and range. Domain Range 25

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27 Power Functions Set the window on your calculator for problems 3 5 to: xmin 2, xmax 2, ymin - 4, ymax 16. Graph all three of the functions on the same graph. 16 1) Graph y = x 4. Sketch and label the function. 2) Graph y = x 8. Sketch and label the function. 3) Graph y = x 12. Sketch and label the function These functions are even power functions. -4 Using the graphs above, answer questions 6 and 7. 4) What do you notice about the domain and range of an even power function? 5) What do you notice about the symmetry of an even power function? Graph the same three power functions above in a new window: xmin -1, xmax 1, ymin 0, ymax 1 6) What happens to the graph of an even power function when the value of the exponent increases? ) Where do the functions intersect? 8) Will y = x 14 intersect at the same points as the given 3 functions? Why or why not? Change the window on your calculator for problems to: xmin 2, xmax 2, ymin -16, ymax 16. Graph all three of the functions on the same graph. 16 9) Graph y = x 3. Sketch and label the function. 10) Graph y = x 7. Sketch and label the function. 11) Graph y = x 11. Sketch and label the function

28 These functions are odd power functions. Using the graphs of the odd power functions, answer questions ) What do you notice about the domain and range of an odd power function? 13) What do you notice about the symmetry of an odd power function? Graph the same three power functions above in a new window: xmin -1, xmax 1, ymin -1, ymax 1 14) What happens to the graph of an odd power function when the magnitude increases? (What happens when the exponent gets bigger?) ) Where do the functions intersect? 16) Will y = x 15 intersect at the same points? Why or why not? -1 17) Graph 5 functions of the form y = ax b, where a > 0, b > 1, x > 0 and y >0. Use the window [0, 5, 1, 0, 5, 1]. What pattern can you find in the graphs of all of these functions? Sketch a generalization. 18) Graph 5 functions of the form y = ax b, where a > 0, x > 0, y > 0, 0 < b < 1. Use the window [0, 5, 1, 0, 5, 1]. What pattern can you find in the graphs of all of these functions? Sketch a generalization. 19) Negative Integer exponents Sketch the graphs of y = x -4 and y = x -8 Sketch the graphs of y = x -3 and y = x -9 28

29 Sketch the end behavior of each function without using a graphing calculator and then describe the end behavior. 1. y = 3x 3 2. y = x y = x y = (x 2) 6 Parent Parent Parent Parent Function: Function: Function: Function: Transformations: Transformations: Transformations: Transformations: Left: x, y Left: x, y Left: x, y Left: x, y Right: x +, y Right: x +, y Right: x +, y Right: x +, y 1. y = 3x 3 2. y = x y = x y = (x 2) 6 Parent Parent Parent Parent Function: Function: Function: Function: Transformations: Transformations: Transformations: Transformations: Left: x, y Left: x, y Left: x, y Left: x, y Right: x +, y Right: x +, y Right: x +, y Right: x +, y 29

30 A. y = 2x + 3 B. y = x C. y = x D. y = x E. y = x F. y = x G. y = x H. y = 2x + 3 I. y = x

31 Modeling with Power Functions: 31

32 Unit 4 Test Review Graph the function y = ceiling x y = 2 x 32

33 7. What is the vertex of the graph of the function y = Ix + 2I 4? 8. Compare the graphs of the pair of functions. Describe how the graph of the second function relates to the graph of the first function. y = -2IxI and y = -2IxI Write the equation that is the translation of y = IxI left 1 unit and up 2 units. 10. Graph the function. 11. Suppose that x and y vary inversely, and x = 7 when y = 11. Write the function that models the inverse variation. 12. Suppose that x and y vary inversely and that y = 1 6 variation and find y when x = 10. when x = 3. Write a function that models the inverse 33

34 13. A drama club is planning a bus trip to New York City to see a Broadway play. The cost per person for the bus rental varies inversely as the number of people going on the trip. It will cost $30 per person if 44 people go on the trip. How much will it cost per person if 60 people go on the trip? Round your answer to the nearest cent, if necessary. Sketch the asymptotes and graph the function Write an equation for the translation of y = 4 that has the asymptotes x = 7 and y = 6. x Solve each equation. Remember to check for extraneous solutions x+5 = 1 x 6 x 2 +x x 2 +x x+1 34

35 Homework Answers Piecewise Functions Practice Homework 1. 2, 1, 0.5, all reals, y , 800, 960, 5350 Taxes paid on: $2500, $16250, $ $40, $40, $70, $0, 0 and if 0 < x 10 f(x) = { (x 10) if 10 < x 50 } #2. Graph in thousands Absolute Value HW 11. y = x y = x h = 0, k =2 21 h = -2, k = y = x y = x 3 Step Function Hw a) 5 b) 4 c) 9 d) -7 e) 0 f) left circles are open a) 0.58 b) 1.75 c) No. Cost $

36 Square root and cube root homework Joint and Combined Variation HW Answers: 1. z = 10/9 2. a = velocity = 30 mph Solving Rational Equations HW 1. 1/ / y = 66/5 5. p = force = dynes 5. -1/ /5 Graphing Simple Rational Equations HW

37 Power Function Graph Matching 1. G 2. B 3. F Modeling with Power Functions 1. c = 81.67x cal 2. a. d=1.7, 3.4, 20.4,102 km d =.34t b. 9.8 mins c. A = 9.1, 36.3, 1307, km^2 d. P = 11.26t 2 4. H 5. A 6. E e. 298 s or approx. 5 min f. 7. I 8. C 9. D 3. a. T = kr 2 D 4 b. increases by factor of 4 c. increases by factor of 16 d. Reduced to 44.4% 4. decreased by % or (70.711% of original V) 37

38 Unit 4 Test Review Answers y = x y = 77/x 12. y = cost = (2,-4) 8. asymptotes y = -3 x = y = + 6 x x = -2 x x = 2, x x = 4, 1 x -1 or 0 2 nd graph is translated down 3 38

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