2-1: Relations and Functions. Mr. Gallo Algebra 2. What is a Relation

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1 -1: Relations and Functions Mr. Gallo Algebra What is a Relation 1

2 In 000, the 4 most populous states(in millions), were CA {3}, TX {1}, NY {19} and FL {16}. The numbers of U.S. Representatives were CA {53}, TX {3}, NY {9} and FL {5}. How can you represent a relation for these data in 4 different ways? Domain and Range of a Relation x y $5.00 $10.00 $15.00 $0.00 $5.00 $30.00 Domain: Range:

3 Identifying Functions ,9,4,,1, 11,9,10 5 Components of Functions 3

4 The Vertical Line Test Why the Vertical Line Test Works 4

5 Homework: Types of Function Notation Euhler s Function Notation: Mapping Notation: 5

6 Euhler s Notation: y f (x) Ex.1: Here, the three distances (thinking, braking, stopping) are each a function of speed. Rewrite as Function Rules using Euhler s f(x). Thinking distance at speed x: y= x. Braking distance at speed x: y x 0 Stopping distance at speed x: x y x 0 6

7 Using Function Notation For f x x, what is the output for the input -4, 0 and 0.? x Input -4 3 Function Rule f x x 3 f x Output 0 0. A pizza costs $14; the flat delivery fee is $1.50. What function rule models the total cost of the number of pizzas delivered? Evaluate for 5 pizzas. 7

8 Complete Got It? p. 64 #6 Complete Lesson Check p. 64 # 1-4 8

9 Mr. Gallo Algebra Ex. 1: In New York, you can receive a refund for returning aluminum cans. For each can returned, you receive a 5 refund. a). If r represents the refund and c represents the number of cans returned, write an equation to show this relationship. b). Use your calculator to graph the equation. Re size the window (y max) and use (trace). What happens to r as c increases? 1

10 The aluminum can problem is an example of a: refund r varies as c the # of cans. When the number of cans, the refund. When the number of cans, the refund. Direct Variation Function: y y kx where k 0 or k where k 0 x Independent Variable: Dependent Variable: k is the 0.5 Ex. : Consider the function rule f x x: 1. Graph the function rule. What does the graph look like?. Use your calculator to make a table. 3. What is the constant of variation, k? f x x x f(x)

11 Write an direct variation equation that describes the following (let k=constant of variation): 1. The cost c of gas varies directly as the amount of gas g purchased.. The price P of breakfast cereal is directly proportional to the number of boxes b of cereal purchased. Identifying Direct Variations from Equations For each function, determine whether y varies directly with x. If so, what is the constant of variation? a). y x b). y 6 x 3

12 Using Proportions to Solve Direct Variations Suppose y varies directly with x, and y=4 when x=5. What is x when y=10? Set up a proportion and solve for the missing value of x: k y x x 4 5 4x x The value of x=6.5 when y=10. Solving Direct Variation Problems: Follow the following four steps: 1. Write an that describes the function.. Solve for the. 3. the variation function using the constant of variation found in step. 4. the function for the desired value of the independent variable. 4

13 Solving Direct Variations Problems Suppose that lightning strikes a known point 4 miles away, and that you hear the thunder 0 seconds later. Then, how far away has lightning struck if 30 seconds pass between the time you see the flash and hear its thunder? Here, the distance varies directly as the time it takes for us to hear the thunder. How do we solve this? Ex. 3: The cost of buying fancy nuts varies directly with the weight. If 8.5 kg of nuts cost $47.60, how much do 0 kg cost? 5

14 8-1:Inverse Variation Mr. Gallo Algebra Ex. 1: It takes one student 36 working hours to wash all of the windows at the school. If more students helped, then each student would work less time (the total working time would still be 36 hours). a). Let t = time each student works and s = number of students washing the windows, write an equation to show this relationship. b). Use your calculator to graph the equation. Re-size the window (y-max) and use (trace). What happens to t as s increases? 1

15 c). Create a table using your calculator and answer the following questions 1. What happens to the time each student works if you double the number of students working?. What happens to the time each student works if you triple the number of students working? 3. What happens to the time each student works if you halve the number of students working? Inverse Variation Function: Independent Variable: Dependent Variable: k is the

16 Write an inverse variation equation that describes the following (let k=constant of variation): 1. The speed s you travel in a car varies inversely with the time t it takes you to get there.. The warmer the temperature t gets on a snowy winter day varies inversely as the amount of snow s left on the ground. Solving Inverse Variations Problems Nancy and Sam are trying to balance the seesaw. The distance d a person sits from the fulcrum is inversely proportional to the person s weight w. Sam is sitting meters from the fulcrum and weighs 55 kilograms. How far should Nancy sit from the fulcrum if she weighs 50 kilograms? How do we solve this? 3

17 Homework: Combined and Joint Variations Combined Variation One quantity varies with quantities. Combined Variation Joint Variation One quantities varies with quantities. Equation Form z varies jointly with x and y. z varies jointly with x and y and inversely with w. z varies directly with x and inversely with the product wy. 4

18 The volume of gas varies directly with its temperature and inversely with pressure. Volume is 100 m 3 when the temperature is 150K and the pressure is 15 lb/cm. What is the volume when the temperature is 50K and the pressure is 0 lb/cm? 5

19 -3:Linear Functions & Slope-Intercept Form Mr. Gallo Algebra Linear Function A function whose graph is a. A solution to a linear function is any that makes the equation true. Slope The ratio of the vertical change to the horizontal change 1

20 Slope-Intercept Form An equation written in the form: What is the equation of the line: 1 10y 3x -4: More About Linear Equations Mr. Gallo Algebra

21 Finding a Linear Equation: I. Given a Point and the Slope: Find an equation of a line that passes through the point ( 7, 1) and has the slope /3 Use the point-slope form: 1) II. Given Two Points: Find an equation of a line that passes through the points (-3, 6) and ( 5, 0) 1) Use the point-slope form: ) 3

22 Example Suppose you know that the formula relating blood pressure and age is linear and that normal systolic blood pressures are 110 for a 0-year-old and 130 for a 60-year-old. Construct a formula in which blood pressure B is a function of age A. 60,130 y y m x x y 130 x 60 1 y 130 x 30 0,110 Return to the equation you wrote to relate systolic blood pressure and age: 1 y 130 x What graph models the situation? Write the equation in Slope-Intercept Form If someone is 30 yrs. old, what is their systolic blood pressure expected to be? Age 4

23 Standard Form of a Linear Equation The Standard Form of a linear equation is: What is an equation of the line integer coefficients. y x 3 in standard form? Use 5 y x 3 5 5

24 Summary of Equation Forms Writing Equations of Lines Example: Graph the equation 10x + 6y = 30 by using its intercepts: 6

25 What is the equation of each line in slope-intercept form? 1. The line parallel to y 5x 4through (-,1). 3. The line perpendicular to y x with the same y-intercept as x 3y

26 -5: Using Linear Models Mr. Gallo Algebra Scatterplots & Correlation Scatterplot Correlation 1.. 1

27 Positive Correlations Negative Correlations No Correlation Complete Got It? #1 p.93 a. Strong negative b. About $170 Linear Regression Function TI Enter the data into STAT EDIT (L 1, L, L 3, etc.). Press STAT; move the cursor to CALC and press 4:LinReg(ax+b) 3. Enter the lists followed by commas; press VARS 4. Move the cursor to Y-VARS; press 1:Function 5. Choose Y 1 6. Press ENTER

28 1. Create a scatterplot of the data using your graphing calculator.. What type of correlation do you see in the graph (positive, negative, or none)? 3. Draw a line of best fit using the LinReg function on your calculator. What is the equation of this line? 4. Use this equation to predict the number of traffic deaths in the United States in There were approximately million cars in the US in Country Cars (Millions) Traffic Deaths Argentina 4.3 3,054 Australia 7.7 4,10 Belgium 3.8 1,937 Bulgaria 1.3 1,409 Canada 1.6 4,10 France ,198 Israel Italy 7.3 8,717 Japan ,398 W. Germany ,435 Correlation Coefficient The number from the linear regression model is called the. This number is from to. The absolute value of the r number indicates the of the relationship. When r =, there is a perfect linear relationship. When r =, there is a no linear relationship. 3

29 Correlation Coefficient The closer the number is to, the stronger the linear relationship between the variables. The closer the number is to, the the linear relationship between the variables. The sign of the r-value indicates a positive or negative slope for the linear regression. In this case the sign is, therefore a slope indicated. Below is the population of Kansas based on census data for the years 1900 through a. Create a scatterplot of the data. b. What type of correlation do you see in the graph (positive, negative, or none)? c. Draw a line of best fit using the LinReg function on your calculator. What is the equation of this line? d. Use this equation to predict the population of Kansas in 010. Year Population (thousands) , , , , , , , , , ,490 4

30 -6:Families of Function & -7: Absolute Value Functions and Graphs Mr. Gallo Algebra Parent Functions & Graph-Translations Parent Function: Graph-Translation (Transformation): 4 4 qx = x + fx = x -5 5 hx = x+3 fx = x gx = x-3 - rx = x - Horizontal Translations Vertical Translations 1

31 Stretches and Compressions Stretch Compression 6 ux = 8 x 4 4 sx = x fx = x 1 vx = x 5 fx = x 5 1 wx = - - x Stretch Compression Combining Transformations The general form of absolute value functions:

32 Use a graphing calculator to create graphs of: Equation Vertex Axis of Symmetry y x y x 3 y y x 3 x 5 Conclusions 1. Subtracting h from x translates the graph Vertex Axis of Symmetry. Adding h to x translates the graph Vertex Axis of Symmetry 3. Subtracting k from y translates the graph Vertex Axis of Symmetry 4. Adding k to y translates the graph Vertex Axis of Symmetry 3

33 Examples: 1. What is the equation of the absolute value function? Step 1: Step : 5 Step 3: 4

34 -8: Graphing Inequalities in the Coordinate Plane Mr. Gallo Algebra Graphing Inequalities in the Coordinate Plane When a line is drawn in a plane, the line separates the plane into distinct regions called. The line itself is the of the two regions. The boundary does belong in either half plane. 1

35 When graphing an inequality, follow these steps: 1. Example 1: Graph y > x When graphing an inequality, follow these steps: 1 Example : Graph y -3x -. 3.

36 Example 3: You put up a new shelf that is 1 ft. wide to store some of your books and trophies. Each book takes up 1 in. and each trophy takes up 3 in. what is a graph showing how many books and how many trophies will fit on the shelf? 4 Trophies Books 4 Trophies Books This is a so the shaded region really represents all the points for the possibilities. 3

37 When graphing an Absolute Value inequality, follow these steps: 1.. Graphing an Absolute Value Inequality Graph y 1 x What Inequality Does This Graph Represent? 4 1. How is the vertex translated?. Is the solution above or below the boundary line? 3. Is the boundary line dashed or solid? 4 4