QUADRATIC FUNCTIONS AND EQUATIONS Student Packet 1: Introduction to Quadratic Functions

Transcription

1 Introduction to Quadratic Functions QUADRATIC FUNCTIONS AND EQUATIONS Student Packet 1: Introduction to Quadratic Functions QUAD1.1 Functions Revisited Define the domain and range of a function. Learn some function notation QUAD1.2 Modeling with Quadratic Functions 1 Create mathematical models of an event. Determine the degree of accuracy appropriate for a given situation. Graph different data sets that result in lines or parabolas. Use technology to display and analyze data. Understand some basic properties of a parabola. QUAD1.3 Creating Parabolas from Lines Graph linear and quadratic functions Explore some basic properties of a quadratic function. QUAD1 STUDENT PACKET QUAD1.4 Vocabulary, Skill Builders, and Review QUAD1 SP0

2 Introduction to Quadratic Functions Word or Phrase function domain of a function linear function parabola vertex of a parabola quadratic function range of a function WORD BANK (QUAD1) Definition or Explanation Example or Picture QUAD1 SP0

3 Introduction to Quadratic Functions 1.1 Functions Revisited Ready (Summary) We will revisit the concept of function and graph of a function. We will define the domain and range of a function, and learn some function notation. Fill in the t-tables and draw the graphs. FUNCTIONS REVISITED Go (Warmup) Set (Goals) Define the domain and range of a function. Learn some function notation 1. y = x x = y 2 1 x y x y NOTE: In this unit, the horizontal axis will always be the x-axis, the vertical axis will always be the y-axis, and the input variable will always be x, unless otherwise noted. QUAD1 SP1

4 Introduction to Quadratic Functions 1.1 Functions Revisited A REVISIT TO FUNCTIONS AND THEIR GRAPHS A function is a rule in which each input value is assigned to exactly one output value. The graph of a function is the set of ordered pairs, each consisting of an input and its corresponding output. If the inputs and outputs are real numbers, then we can represent the graph of a function as points on the coordinate plane. The vertical line test refers to a visual way to determine whether a set of ordered pairs in the usual xy-plane is the graph of a function. The vertical line test simply states that if a vertical line intersects a graph in more than one point, then it is a NOT a function. 1. Consider the equation and its graph in problem 1 of the warmup. Why does it represent a function of x based upon: a. the definition of function? b. the vertical line test? c. the equation? 2. Consider the equation and it s graph in problem 2 of the warmup. Why does it not represent a function of x based upon: a. the definition of function? b. the vertical line test? c. the equation? QUAD1 SP2

5 Introduction to Quadratic Functions 1.1 Functions Revisited DOMAIN AND RANGE The domain of a function is the set of inputs of the function. The range of a function is the set of outputs of the function. Consider this set of ordered pairs: {(1, 10), (2, 20), (3, 30), (4, 40)} 1. Do these ordered pairs represent a function? Explain.. 2. The domain is {,,, }. 3. The range is {,,, } 4. What would be an ordered pair that, if added to the set, would make the relationship NOT a function? Consider this graph, which consists of ordered pairs in the form (x, y). From smallest to largest: 3. The x-coordinates include all real values from to. We can write this as the closed interval: [, ] We can write this as an inequality: x 4. The y-coordinates include all real values from to. We can write this as a closed interval: [, ] We can write this as an inequality: y 5. Does this graph represent a function of x? If it is a function, what is the domain? the range? QUAD1 SP3

6 Introduction to Quadratic Functions 1.1 Functions Revisited DOMAIN AND RANGE (continued) Consider this graph, which consists of ordered pairs in the form (x, y). Assume that the lines continue outside the graphing area to the left and right with no surprises, as indicated by the arrows. From smallest to largest: 6. The x-coordinates include all real values from to. We can write this as an open interval: (, ) We can write this as an inequality: < x < 7. The y-coordinates include all real values from to. We can write this interval: (, ] We can write this as an inequality: < y 8. Does this graph represent a function of x? If it is a function, what is the domain? the range? Consider this graph, which consists of ordered pairs in the form (x, y). Assume that the entire graph is shown here. 9. The x-coordinates include all real values from to (but not including ). Write this using interval notation: Write this as an inequality: 10. The y-coordinates include all real values from to. Write this using interval notation: Write this as an inequality: 11. Does this graph represent a function of x? If it is a function, what is the domain? the range? QUAD1 SP4

7 Introduction to Quadratic Functions 1.1 Functions Revisited FUNCTION NOTATION The notation f(x) (read f of x ) is used to denote a function in which x is an element of the domain (the input value). Therefore, f(x) is the output of the function f that corresponds to the input x. Error alert: f(x) does NOT mean f multiplied by x. Example of a linear function: y = 3x + 2 Example of a linear function using the new notation: f(x) = 3x + 2. To evaluate this function at x = 5, we can simply write: f(5) = 3(5) + 2 = 17 Given function: g(x) = x 2 7 Evaluate the function g for the given inputs 1. g(3) = 2. g(1) = 3. g(0) = 4. g(-2) = Given function: h(x) = 2x 2 3x + 1 Evaluate the function h for given inputs 5. h(3) = 6. h(0) = 7. h(-2) = 8. challenge: h(a) = 9. Graph the function f(x) = x 2 2x + 1. First make a table. Write the ordered pair this represents Write the ordered pair this represents QUAD1 SP5

8 Introduction to Quadratic Functions 1.2 Modeling with Quadratic Functions 1 MODELING WITH QUADRATIC FUNCTIONS 1 Ready (Summary) We will conduct an experiment with a flashlight and collect data. We will use multiple representations to organize, display, and analyze data. We will create mathematical models to fit our data. Set (Goals) Create mathematical models of an event. Determine the degree of accuracy appropriate for a given situation. Graph different data sets that result in lines or parabolas. Use technology to display and analyze data. Understand some basic properties of a parabola. Go (Warmup) A function is a rule that assigns to each input x a unique output y = f(x). Fill in the t-table and draw the graph for each function. Use a different color for each graph. Not all values will fit on the graph x y = x 2 y = x y = 2x color 4. How are the graphs for #1 and #2 the same? How are they different? 5. How are the graphs for #2 and #3 the same? How are they different? QUAD1 SP6

9 Introduction to Quadratic Functions 1.2 Modeling with Quadratic Functions 1 FLASHLIGHT INVESTIGATION: RECORDING PAGE Your teacher will perform a flashlight demonstration. 1. Write your observations about what quantities are changing and how you might measure those changes. 2. You will design and conduct a flashlight investigation. Record the data you collect on this page. Write a short description of this investigation. Underline variables in your description (what you are measuring). What is a reasonable degree of accuracy for your measurements and calculations? Label columns as needed and record your data here. 4. Perform other flashlight investigations (or repeat the one you just did), and collect additional data. Record results on blank paper. QUAD1 SP7

10 Introduction to Quadratic Functions 1.2 Modeling with Quadratic Functions 1 FLASHLIGHT INVESTIGATION: A LINEAR FUNCTION Compare two variables from your experiment that you think will result in a linear function. Record your data in the table, or use data provided by your teacher. 1. Briefly describe the experiment that was performed to gather the data. 2. What variables are being compared? 3. What will go on the horizontal axis? How will you scale it? 4. What will go on the vertical axis? How will you scale it? 5. What is a good title for your graph? 6. What does the y-intercept on your graph represent? 7. Estimate a line of best fit for your graph QUAD1 SP8

11 Introduction to Quadratic Functions 1.2 Modeling with Quadratic Functions 1 FLASHLIGHT INVESTIGATION: A QUADRATIC FUNCTION Compare a variable that represents a length measurement to a variable that represents an area measurement. Record your data in the table, or use data provided by your teacher. 1. Briefly describe the experiment that was performed to gather the data. 2. What variables are being compared? 3. What will go on the horizontal axis? How will you scale it? 4. What will go on the vertical axis? How will you scale it? 5. What is a good title for your graph? 6. What does the y-intercept on your graph represent? 7. What is the shape of your graph? QUAD1 SP9

12 Introduction to Quadratic Functions 1.2 Modeling with Quadratic Functions 1 FINDING AN EQUATION FOR THE DATA The graph of a quadratic function is a curve called a parabola. Functions whose graphs are parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. Parabolas that have symmetry about the y-axis have the form: y = ax 2 + k This parabola has its vertex at (0, k). The value of a determines its steepness. We will estimate a quadratic function to fit the data on the previous page. 1. Estimate a value for k. The minimum point on the graph is (0, ). This coordinate is called the vertex of the parabola. Use this as a value for k. y = ax Substitute (x, y) coordinates from the graph on the previous page to calculate some possible values for a. For (, ) For (, ) For (, ) y = ax 2 + k = (a)( ) 2 + a = y = ax 2 + k = (a)( ) 2 + y = ax 2 + k For (, ) For (, ) For (, ) QUAD1 SP10

13 Introduction to Quadratic Functions 1.2 Modeling with Quadratic Functions 1 FINDING AN EQUATION FOR THE DATA (continued) 3. Describe a strategy for estimating a value for a. 4. Use your strategy to estimate a value for a. 5. Using your calculations and estimations, write a quadratic function to fit your data y = ax 2 + k y = x Locate a program on the internet, or use a graphing calculator that finds a quadratic function to best fits a set of data. Find the equation of a parabola that best fits your data. 8. Compare your estimated quadratic function to the one generated by technology. Discuss discrepancies. 9. Locate a program on the internet or use a graphing calculator that finds a linear function to best fit a set of data. Find the equation of a line that best fits the linear function you created from flashlight investigation data. 10. Compare your estimated linear function to the one generated by technology. Discuss discrepancies. QUAD1 SP11

14 Introduction to Quadratic Functions 1.3 Creating Parabolas from Lines Ready (Summary) CREATING PARABOLAS FROM LINES By multiplying two linear expressions, we will generate a quadratic expression that defines a quadratic function. Graphically, we will observe that multiplying the y- coordinates of the two linear functions yields the y-coordinate of the quadratic function. We will learn some basic properties of parabolas. Go (Warmup) Set (Goals) 1. Use two different colored pencils. Graph the following lines: f(x) = 2x + 2 g(x) = x 3 x f(x) x g(x) Graph h(x) = f(x) g(x) by following your teacher s directions. Graph linear and quadratic functions Explore some basic properties of a quadratic function. QUAD1 SP12

15 Introduction to Quadratic Functions 1.3 Creating Parabolas from Lines PRODUCTS OF LINEAR EXPRESSIONS Consider the two lines you graphed in the warmup: f(x) = 2x + 2 and g(x) = x Find the product h(x) = f(x) g(x). That is, h(x) = (2x + 2)(x 3) = 2. Fill in the in the table. x f(x) = g(x) = h(x) = Write ordered pairs for the intercepts for f(x) and g(x). f(x) = g(x) = h(x) = x-intercept (, ) (, ) (, ) (, ) y-intercept (, ) (, ) (, ) 4. Compare the ordered pairs for (x, h(x)) to the points graphed for the parabola on the warmup page. What do you notice? 5. How do the x-intercepts of the lines compare to the x-intercept(s) of the parabola? 6. How do the y-intercepts of the lines compare to the y-intercept(s) of the parabola? 7. Does the parabola have a minimum point or maximum point? What are its coordinates? QUAD1 SP13

16 Introduction to Quadratic Functions 1.3 Creating Parabolas from Lines CREATING PARABOLAS FROM LINES 1 1. Graph the lines f(x) = x and g(x) = -x + 6 using two different colors. 2. Find the intercepts for each function. f(x) = g(x) = x-intercept (, ) (, ) y-intercept (, ) (, ) 3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color. 4. Find an equation for the parabola. h(x) = f(x) g(x) = 5. Identify the following coordinates for the parabola: a. x-intercept(s) b. y-intercept(s) c. vertex 6. Is the vertex a minimum or a maximum point? 7. Where is the vertex located in relation to the x-intercepts? QUAD1 SP14

17 Introduction to Quadratic Functions 1.3 Creating Parabolas from Lines CREATING PARABOLAS FROM LINES 2 1. Graph the lines f(x) = -2x 4 and g(x)= -x 2 using two different colors. 2. Find the intercepts for each function. f(x) = g(x) = x-intercept (, ) (, ) y-intercept (, ) (, ) 3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color. 4. Find an equation for the parabola. h(x) = f(x) g(x) = 5. Identify the following coordinates for the parabola: a. x-intercept(s) b. y-intercept(s) c. vertex 6. Is the vertex a minimum or a maximum point? 7. Where is the vertex located in relation to the x-intercepts? QUAD1 SP15

18 Introduction to Quadratic Functions 1.3 Creating Parabolas from Lines CREATING PARABOLAS FROM LINES 3 1. Graph the lines f(x) = 2x and g(x) = -x + 1 using two different colors. 2. Find the intercepts for each function. f(x) = g(x) = x-intercept (, ) (, ) y-intercept (, ) (, ) 3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color. 4. Find an equation for the parabola. h(x) = f(x) g(x) = 5. Identify the following coordinates for the parabola: a. x-intercept(s) b. y-intercept(s) c. vertex 6. Is the vertex a minimum or a maximum point? 7. Where is the vertex located in relation to the x-intercepts? QUAD1 SP16

19 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review FOCUS ON VOCABULARY (QUAD1) Complete the flow chart to show how this week s vocabulary is related. 1. A rule where each input matches exactly one output 2a. The set of input values for a function 2b. The set of output values for a function 3. The type of function described by y = 2x The type of function described by y = 2x The shape of y = 2x + 3 (a word not on the vocabulary list) 6. The shape of y = 2x a. _ 2b QUAD1 SP17

20 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 1 Multiply: 1. (x + 3) (x 2) 2. (x + 6) (x 6) 3. (-2x - 1) (3x + 7) 4. (5x + 3) (5x + 3) 5. (2x - 5) (3x) (x + 1) Factor: 7. 3x x x 2 8x x 2 + 7x x x 2 15x + 18 Evaluate each expression when a = 1, b = 4, c = -2, and x = 1 2 : 13. ax 2 + c 14. (5c - 2) (5c + 2) 16. 2ax 2 bx + c 17. ac + b a b 2 b 4ac 2a QUAD1 SP18

21 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 2 1. Between which two consecutive integers is: a. 18? and. b. - 18? and. c ? and. d ? and. e. 2-18? and. f ? and. Simplify each radical expression ( ) ( ) QUAD1 SP19

22 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 3 Which of the following input-output tables illustrate functions? If it is not a function, state why not. 1. x (input) y (output) x (input) y (output) Which of the following sets of ordered pairs illustrate functions? If it is not a function, state why not. 3. (1,5), (2,6), (3,5), (4,6) 4. (10,8), (20,16), (30,24), (8,10) 5. (1.5,5), (2.8,12), (1.5,7), (2.8,15) Which of the following mapping diagrams illustrate functions? If it is not a function, state why not QUAD1 SP20

23 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 4 Fill in the t-table and draw the graph for each equation. Use a different color for each graph x y = x y = x + 3 y= 2x color Solve each inequality. Then draw a number line and graph the solution. 4. 2x + 1 < x + 1 < -7 Graph: x 28 Graph: x 28 Graph: Graph: QUAD1 SP21

24 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 5 Fill in the t-table and draw the graph for each equation. Use a different color for each graph x y = -x y = -x + 1 y = -2x color 4. Evaluate 4x 2 + 2x 3xy + 5y 6y 2 for x = -5, y = 10. Multiply. 5. (x + 3)(x + 7) 6. (x + 3)(x 7) 7. (x 3)(x + 7) 8. (x 3)(x 7) QUAD1 SP22

25 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 6 On each graph below, the x-axis is the horizontal axis and the y-axis is the vertical axis. Which of the following graphs illustrate functions of x? If it is not a function, state why not. If it is a function, state the domain and range Sketch a nonlinear graph that represents a function 6. Sketch a linear graph that does NOT represent a function QUAD1 SP23

26 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 7 Factor. 1. 3x 2 + 2x x 2 + 5x x 2 + 4x x 2 4x x 2 4x x 2 10x x 2 + 4x 8. 10x 2 15 Evaluate if a = 15, b = -1, and c = b 4ac 2 b 4ac 10. b b 2a Evaluate if a = 2, b = 0, and c = b 4ac 2 b 4ac 13. b b 2a 2 b 4ac 2a 2 b 4ac 2a QUAD1 SP24

27 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 8 Fill in the t-table and draw the graph for each equation. Use a different color for each graph x f(x) = x 2 g(x) = x 2 3 h(x) = 2x color 4. In order to ride a rollercoaster, all riders must be at least 60 inches tall. Using h to represent the height, write an inequality that shows the height of all the riders on the rollercoaster. Then draw a number line and graph the solution. Inequality: Graph: 5. In order to ride on the ponies at a fair, all riders must weigh less than 50 pounds. Using w to represent the weight, write an inequality that shows the weight of all the riders on the ponies. Then draw a number line and graph the solution. Inequality: Graph: QUAD1 SP25

28 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER 9 Fill in the t-table and draw the graph for each equation. Use a different color for each graph x f(x) = -x 2 g(x) = -x h(x) = -2x color 4. Draw a coordinate plane on the graph below and then graph the following system of linear inequalities. Don t forget to shade the region that represents all the coordinates that are solutions to the system. y < 2x +1 y x 3 QUAD1 SP26

29 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 SKILL BUILDER Graph the lines f(x) = 1 x + 3 and g(x) = 1 x 3 using two different colors Find the intercepts for each function. f(x) = g(x) = x-intercept (, ) (, ) y-intercept (, ) (, ) 3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color. 4. Find an equation for the parabola. h(x) = f(x) g(x) = 5. Identify the following coordinates for the parabola: a. x-intercept(s) b. y-intercept(s) c. vertex 8. Is the vertex a minimum or a maximum point? 9. Where is the vertex located in relation to the x-intercepts? QUAD1 SP27

30 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 TEST PREPARATION (QUAD 1) Show your work on a separate sheet of paper and choose the best answer. 1. What is the domain of the set {(-4, 1), (1, -2), (-1, -3)} A. {-4, 1, -1} B. {1, -2, -3} C. {-4, 1, -2, -1, 3} D. Cannot determine 2. Is the set of ordered pairs {(-4, 1), (1, -2), (1, -3)} a function? A. No B. Yes C. There aren t enough points to tell D. Can t tell without a coordinate graph 3. How do you read the function notation f(x) A. f multiplied by x B. f of x C. f parentheses x D. f before x 4. Evaluate f(x) = b 2 4ac if a = -1, b = 4, c = 2 and write in simplest form. A. 2 2 B. 2 6 C What are the x-intercepts of the function h(x) = (x + 3) (x + 2) A. 3 and 2 B. 6 C. -3 and -2 D. D. Not enough information to evaluate Cannot tell from the function. 6. Geanie is collecting data about her commute to school. What variable is probably NOT relevant? A. Distance from school to home B. Geanie s walking pace C. Time it takes to get to school D. Geanie s height QUAD1 SP28

31 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 KNOWLEDGE CHECK (QUAD1) Show your work on a separate sheet of paper and write your answers on this page. QUAD1.1 Functions Revisited Refer to the graph at the right. 1. Does the graph represent a function of x? Explain how you know. 2. If it is a function, state the domain and range. QUAD 1.2 Modeling with Quadratic Functions 1 3. Matty put a pencil on a projector and then moved the projector away from the wall. What do you think will happen as he moves the projector? What are two variables that Matty might measure? Do you think the variables you selected are related? If so, how? 4. Graph y = x and y = x 2 3. How are these graphs the same? How are these graphs different? QUAD 1.3 Creating Parabolas From Lines For the function f(x) = (x 1)(x): 5. Name the x-intercept(s). 6. Name the y-intercept(s). 7. Find the vertex. 8. Graph the function. 9. Describe the shape of the function. QUAD1 SP29

32 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 This page is left blank intentionally. QUAD1 SP30

33 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 This page is left blank intentionally. QUAD1 SP31

34 Introduction to Quadratic Functions 1.4 Vocabulary, Skill builders, and Review 1.3 This page is left blank intentionally. QUAD1 SP32

35 Introduction to Quadratic Functions HOME-SCHOOL CONNECTION (QUAD1) Here are some questions to review with your young mathematician. For the function g(x) = (x 1)(x + 7) 1. Graph the function. 2. Name the x-intercept(s). 3. Name the y-intercept(s). 4. Find the vertex. 5. Graph the function. 6. State the domain 7. State the range. Parent (or Guardian) signature QUAD1 SP33

36 Introduction to Quadratic Functions 8.F.1 8.F.4 A-APR-3 A-CED-2 A-REI-10 F-IF-1 F-IF-7a MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 SELECTED COMMON CORE MATHEMATICS STANDARDS Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases: Graph linear and quadratic functions and show intercepts, maxima, and minima. STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. First printing copyright 2012 QUAD1 SP34