Foundations of Math II

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1 Foundations of Math II Unit 6b: Toolkit Functions Academics High School Mathematics

2 6.6 Warm Up: Review Graphing Linear, Exponential, and Quadratic Functions 2

3 6.6 Lesson Handout: Linear, Exponential, and Quadratic Functions Is this graph increasing or decreasing? How do you know? What is the domain and range of this graph? How do you know? 3

4 What is the domain and range of this graph? How do you know? 4

5 What is the domain and range of this graph? How do you know? 5

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8 Exploring Exponential Functions 8

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10 6.7 Lesson Handout: Graphing Power Functions Use a table of values to sketch a graph of all three of the functions on the same graph. Use your calculator to verify the graph. Set your window to the following values: xmin -2, xmax 2, ymin -2, ymax 2 1) Graph y = x 4. Sketch and label the function. 2) Graph y = x 6. Sketch and label the function. 3) Graph y = x 10. Sketch and label the function. X x 4 x 6 x These functions are even power functions. Using the graphs above, answer questions ) What do you notice about the domain and range of the parent functions of EVEN POWER FUNCTIONS? 5) What do you notice about the symmetry of an EVEN POWER FUNCTION? 6) What do you notice about the end behavior of an EVEN POWER FUNCTION? Graph the same three power functions above in a new window: xmin -1, xmax 1, ymin -1, ymax 1 7) What happens to the graph of an EVEN POWER FUNCTION when the exponent gets bigger? 8) Where do the functions intersect? 9) Will y = x 14 intersect at the same points? Why or why not? 10

11 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. 10) Graph y = x 3. Sketch and label the function. 11) Graph y = x 7. Sketch and label the function. 12) Graph y = x 11. Sketch and label the function. X x 3 x 7 x 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 the parent functions of ODD POWER FUNCTIONS? 14) What do you notice about the symmetry of an ODD POWER FUNCTION? 15) What do you notice about the end behavior of an ODD POWER FUNCTION? Graph the same three power functions above in a new window: xmin -1, xmax 1, ymin -1, ymax 1 16) What happens to the graph of an odd POWER FUNCTION when the exponent gets bigger? 17) Where do the functions intersect? 18) Will y = x 15 intersect at the same points? Why or why not? In many applications, the relation that exists between two variables x and y follows a function of the form: y = ax b, where a and b are real numbers > 0. This is called the GENERALIZED POWER FUNCTION. 11

12 6.7 Homework: Graphing Power Functions 12

13 6.8 Practice: Graphing Square Root and Cube Root Functions 13

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15 6.9 Lesson Handout: Graphing Inverse Variation Functions Inverse Variation Why is this called an inverse variation? What is the end behavior of this function? 15

16 Inverse Variation 16

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19 6.9 Homework: Graphing Inverse Variation Functions Graph the function. State the domain, range, the x- and y-intercept(s) and the transformation. 19

20 6.10 Warm Up: Graphing Absolute Value Functions A. (Review): Graph and label intercepts (using ordered pairs) on the graph. 1. y x 2. f(x) x3 3. g(x) x3 x-intercept: x-intercept: x-intercept: B. Explain your knowledge about absolute value using words. C. Using past knowledge to create new knowledge, try graphing the following function: 4. y x Explain your reasoning for the graph you created. Please justify this method (using another method). 20

21 6.10 Lesson Handout: Graphing Absolute Value Functions 21

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26 6.10 Homework: Graphing Absolute Value Functions Graph each of the following functions using a table of values. Then, for each of the functions, state the domain, range, intervals of increasing and decreasing, end behavior, minimum or maximum value, vertex, x-intercept(s), and y-intercept. 1. y x f ( x) x 3 3. g( x) 3 x 6 4. y 2 x f ( x) x g( x) x

27 6.11 Warm Up: Graphing Piecewise Functions Absolute value functions can be written without absolute value if we separate the function into two different functions. Let s explore the function y = x. 1. Sketch the graph of y = x 2. Where in the domain do you think the graph will change from one function to the next? 3. Break the function apart by writing it as two different functions. Include the domain for each part: f ( x) if if x x 27

28 6.11 Lesson Handout #1: Graphing Piecewise Functions 28

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31 6.11 Lesson Handout #2: Graphing Piecewise Functions 1) Make a table of values for the function y = 2x 1 including the points x = -3, -2, -1, 0, 1, and 2. Graph the function y = 2x 1 on the first graph, making sure to graph the points to the right exactly. 2) Make a table of values for the function y = -x + 3 including the points x = 0, 1, 2, 3, 4, 5, and 6. On the second graph, graph the function y = -x + 3, making sure to graph the points to the right exactly. x x y y 3) On the first graph, draw two vertical lines: x = -2 and x = 1. Cut your graph out. Then cut along the vertical lines you drew at x = -2 and x = 1, creating three pieces. Place the piece between x = -2 and x = 1 (the middle piece) on the blank graph below and glue it so that you have the graph of y = 2x 1 between x = -2 and x = 1. Discard the remainder of the graph. 4) On the second graph, draw two vertical lines: x = 1 and x = 5. Cut your graph out. Then cut along the vertical lines you drew at x = 1 and x = 5, creating three pieces. Place the piece between x = 1 and x = 5 (the middle piece) on the blank graph below and glue it on the same graph that has the piece of the graph y = 2x 1 so that you have y = -x + 3 from x = 1 to x = 5. Discard the remainder of the graph. 5) On the graph below, you have graphed the piecewise function ì ï 2x - 1, if -2 x < 1 f(x) = í ï ïî - x + 3, if 1 x 5 31

32 6) On a new set of graphs, go through the same process you just did with the piecewise function ì 3x, if 0 <x 2 f(x) = ï í 6, if 2 <x 4. ï ïî -x + 10, if 4 <x 6 f(x) = 3x f(x) = 6 f(x) = -x + 10 x y X y x y 7) On a new set of graphs, repeat the process one more time with the piecewise function ì - 2x + 4, if x < 0 f(x) = ï í 1. ï x + 4,if 0 x 3 ïî 2 f(x) = -2x + 4 f(x) = 1 2 x + 4 x y x y 32

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35 6.11 Practice: Graphing Piecewise Functions Part I Carefully graph each of the following. Identify whether or not he graph is a function. Then, evaluate the graph at any specified domain value. You may use your calculators to help you graph, but you must sketch it carefully on the grid! f x 1. 2 x 5 x 2 x 2x 3 x 2 Function? Yes or No f f f f x x 1 x 1 x 3 x 1 Function? Yes or No f f f

36 3. f x 2x 1 x 2 5x 4 x 2 Function? Yes or No f f f x x 1 0 f x 2x 1 0 x 5 3 x 5 Function? Yes or No f f f

37 5. f x x 2 x 0 2 x x 4 0 Function? Yes or No f f f f x 5 x 3 2x 3 x 3 Function? Yes or No f f f

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39 6.11 Homework: Graphing Piecewise Functions Evaluate: 1. 3 x, x1 f( x) 2 x, x , x 0 f( x) x, x 0 f (0) f (1) f (2.5) f ( 1) f (0) f (5) 3. 1, x 0 f( x) x 3 x, x x, x1 3 3 f ( x) x, 1 x x 3, x 3 f ( 1) f (0) f ( ) f (.5) f (1) f (3) f (4) 5. 1, x 5 f( x) 0, x x, x 0 3 f ( x) x, 0 x 1 2x1, x1 f (0) f (6) f (5) f ( 1) f (1) f (0) f (2.5) 39

40 Sketch each function below without using a graphing calculator. Find the domain and range of each function. Remember, all functions must pass the vertical line test. 7. x3, x1 f( x) 2 x, x 1 D R f f f (0) f (1) f (2) 8. 2, x 5 f ( x) 2 x, 2 x x, x 2 D R f f evaluate: f ( 2) f (5) 9. x3, x1 f( x) x, x 0 D R f f evaluate: f (1) f (6) f (0) 40

41 10. 2x 3, x 1 f ( x) x 5, 1 x 2 1, x 3 D R f f evaluate: f (1) f (6) f (0) 11. x, 4 x 2 f ( x) x 3, 2 x 1 2 x 2, x1 D R f f evaluate: f ( 4) f ( 2) f (1) 41

42 6.12 Lesson Handout: Graphing Trigonometric Functions Intercepts of the function? Intercepts: 42

43 Intercepts of the function? Intercepts: 43

44 Intercepts of the function? Intercepts: 44

45 6.12 Homework: Graphing Trigonometric Functions 1. Use your calculator to fill in the table below with the values of sine and cosine for each angle. Give each answer to the nearest tenth: sin cos What is the lowest value of sin ; the highest value of sin? What is the lowest value of cos ; the highest value of cos? What will happen if you continue the table past 360? 45

46 2. Use your calculator to fill in the table below with the values of tangent for each angle. Give teach answer to the nearest tenth: tan tan For what values of is tangent undefined? What happens as approaches these values? 46

47 3. What happens after = 360? Can be less than 0? What is the domain of the function y = sin? What is the range of the function y = sin? What is the amplitude of the function y = sin? What are the intercepts of the function y = sin? What is the amplitude of the function y = sin? 47

48 4. How does the graph of y = cos compare to the graph of y = sin? What is the domain of the function y = cos? What is the range of the function y = cos? What is the amplitude of the function y = cos? What are the intercepts of the function y = cos? What is the amplitude of the function y = cos? 48

5. Plot the points for y = tan from the table. What happens when tangent is undefined? How often does this happen? Is there a limit to how large tangent can be?

49 5. Plot the points for y = tan from the table. What happens when tangent is undefined? How often does this happen? Is there a limit to how large tangent can be? What is the domain of the function y = tan? What is the range of the function y = tan? What is the amplitude of the function y = tan? What are the intercepts of the function y = tan? What is the amplitude of the function y = tan? 49

50 6.13 Product Handout: Graphing Toolkit Functions FUNC-Y ART Task: To create a piece of artwork using transformations of toolkit functions. Product: 1. A piece of artwork based upon transformed toolkit functions. 2. A piece of paper on which the equations are listed of the functions used. Write the equations so that the transformations are visible and include the domain and range of each function. Give the viewer some way to know which equation goes with each part in the picture. 3. Include lightly drawn reference axes that show scale. 4. A written synopsis of the process you went through to get the functions in the correct place. It should include struggles and successes that you experienced. Requirements: 1. You must use at least five different toolkit functions. At most two can come from constant, linear, quadratic, and/or cubic. (You can have more than 2 from this set, but only two can count, so at least three have to come from the other functions). 2. You must use each of the following transformations at least once: left, right, up, down, compress, stretch, reflect over the x-axis. 3. You must give a way for the viewer to clearly see which function goes with which part in the final product. Artistic ability is not a component of the evaluation. The goal is to demonstrate that you are able to manipulate a toolkit function to get the shape you want and in the correct place. Create the art in any medium, but needs to be on something more than a regular sheet of paper. Color your final product. Background or auxiliary objects are fine as long as they do not distract in the transformed functions as the focus of the work. 50

51 Place your letter answer in the correct blank below

6.13 Homework: Graphing Toolkit Functions Match the

Names: A) absolute value B) power C) linear D)

x 2 H) y = x 3 I) y = x J) y x 1. 2. 3. 4. 5.

effect of k Identify the parent function name and

52 6.13 Homework: Graphing Toolkit Functions Match the name & equation to the graph. Names: A) absolute value B) power C) linear D) quadratic E) square root Equations: F) y = x G) y = x 2 H) y = x 3 I) y = x J) y x Given: y = a(x-h) 2 +k 11) describe the effect of a on the graph. 12) describe the effect of h on the graph. 13) describe the effect of k on the graph. Identify the parent function name and describe the transformation for each function. 6. g(x) = 3(x-1) 2 6 Parent Function: Transformations: 1) 2) 3) 52

53 1 7. f(x) = - 3 Parent Function: x Transformations: 1) 2) 8. h(x) = 2 x 6 Parent Function: 3 Transformations: 1) 2) 9. f(x) = 3 x 2 3 Parent Function: Transformations: 1) 2) 3) 10. What is the effect on the graph of the function y 2 x 2 when it is changed to y 2 x 3? Name the Parent Function. List the transformations. Graph each equation. Then state the domain, range, x- intercept(s), and y-intercept. 11. y x y 2 x y = - 4x y x 5 53

54 6.14 Review: Key Features of Parent Functions (F-IF.4) Directions: Fill in the characteristics listed under each toolkit function. If a key feature does not apply, write none. Under other attributes, think about things that distinguish that function from others, such as a quadratic function s having a vertex. Also, sketch a graph of each function on the set of axes provided. Linear Function f(x) = x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: Quadratic Function f(x) = x 2 Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: 54

55 Exponential Function f(x) = 2 x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: Power Function: odd f(x) = x odd Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: 55

56 Power Function: even f(x) = x even Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: Square Root Function f(x) = x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: 56

57 Cube Room Function 3 f(x) = x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: Simple Inverse Function f(x) = 1 x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: 57

Absolute Value Function f(x) = x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive:

58 Absolute Value Function f(x) = x Domain: Range: x-intercept(s): y-intercept: Interval where function is increasing: Interval where function is decreasing: Interval where function is positive: Interval where function is negative: Relative max/min: Symmetry: End behavior: As x, y As x, y Other attributes: Sine Function y = sin Domain: Range: x-intercept(s): y-intercept: Period Midline: Amplitude: Other attributes: 58

59 Cosine Function y = cos Domain: Range: x-intercept(s): y-intercept: Period Midline: Amplitude: Other attributes: Tangent Function y = tan Domain: Range: x-intercept(s): y-intercept: Period Midline: Amplitude: Other attributes: 59

Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1

Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the