NOTES: Chapter 6 Linear Functions

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1 NOTES: Chapter 6 Linear Functions Algebra 1-1 COLYER Fall 2016 Student Name:

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3 Section 6.1 ~ Rate of Change and Slope Rate of Change: A number that allows you to see the relationship between two quantities that are changing. FORMULA: Example 1: Finding Rate of Change Using a Table Is the rate of change for each pair of consecutive days the same? What does the rate of change represent? Example 2: Finding Rate of Change Using a Graph The graph shows the altitude of an airplane as it comes in for landing. Find the rate of change. Explain what this rate of change means. Page 3

4 Slope: A number that represents the of a line. FORMULA: Example 3: Finding Slope Using a Graph Find the slope of each line. a. b. Example 4: Finding Slope Using Points Find the slope of the line through A(-2, 1) and B(6, 7). Example 5: Horizontal and Vertical Lines Find the slope of each line. a. b. Page 4

5 REVIEW OF METHODS TO FIND SLOPE: 1) 2) 3) TYPES OF SLOPES FOR DIFFERENT LINES: Use the space below to complete pg #1-4, 7-11, 22-25, 27-29, 31, 32, 40 Page 5

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7 Section 6.2 ~ Slope Intercept Form Linear Function: Parent Function: Linear Parent Function: Linear Equation: The equation of a line gives important information about its graph. Consider the table and graph of the equation y 2x 1. x -2x + 1 y What information do you have about the graph of y 2x 1 from the table? Graph y 2x 1: Page 7

8 y-intercept: Slope-intercept Form of a Linear Equation: The slope-intercept form of a linear equation is y = mx + b. Example 1: Identifying Slope and y-intercept What are the slope and y-intercept of y = 3x 5? Example 2: Writing an Equation Write an equation of the line with slope 3 8 and y-intercept 6. Example 3: Writing an Equation From a Graph Write the equation that models the linear function shown in the graph. Page 8

9 Example 4: Graphing Equations Graph y = 3x 1 Example 5: The base pay of a water-delivery person is $210 per week. He also earns 20% commission on any sale he makes. a. Write an equation to represent the total earnings of the delivery person. b. Graph the equation. Page 9

10 Use this space to complete pg 320 #5-8, 20-24, 34-37, 41-46, 50-51, 56, Page 10

11 Section 6.3 ~ Applying Linear Functions Example 1: A car dealership has 40 cars in stock. The auto manufacturer will deliver new cars to the dealership by car carrier. Each carrier holds 6 cars. a. Write a linear function that relates the number of carriers used to the total number of cars at the dealership. b. Graph the function that models the situation. Page 11

12 Example 2: Analyzing Linear Graphs Students in a ninth-grade class drew the following graph to represent how much money would be in the class fund after washing cars at a fundraiser. a. What does the slope and y-intercept of the graph mean for the given situation? b. If the graph had the same slope but a y-intercept of 15, what could you conclude about the fundraiser? c. If the graph had a slope of 5, what could you conclude about the fundraiser? Page 12

13 Section 6.4 ~ Standard Form Standard Form of a Linear Equation: The standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. x-intercept: The point where a line crosses the. To graph a linear equation in standard form: 1. find the x-intercept by: 2. find the y-intercept by: Example 1: Finding x- and y-intercepts Find the x- and y-intercepts of 3x + 4y = 8. Example 2: Graphing Lines Using Intercepts Graph 2x + 3y = 12 using intercepts. Page 13

14 Example 3: Graphing Horizontal and Vertical Lines a. Graph y = 3. b. Graph x = 2. Example 4: Transforming to Standard Form Write 3 y x 2 in standard form using integers. 4 Page 14

15 Example 5: Write an equation in standard form to find the minutes someone who weighs 150 lb would need to bicycle and swim laps in order to burn 300 calories using the data below. Use this space to complete pg 333 #1-4, 10-12, 14-15, 17-18, 19-22, 23-26, 28-30, 33, 35, 36-37, 62 Page 15

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17 Section 6.5 ~ Point-Slope Form and Writing Linear Equations Point-Slope Form of a Linear Equation: The point-slope form of the equation of a nonvertical line that passes through point ( x, y ) and has slope m is 1 1 Example 1: Graphing Using Point-Slope Form Graph the equation 1 y 5 ( x 2). 2 Example 2: Writing an Equation in Point-Slope form Write the equation of the line that has slope -3 that passes through the point (-1, 7). Page 17

Example 4: Writing an Equation Using a Table Is the relationship shown by the data

18 Example 3: Using Two Points to Write an Equation Write the equations for the line in point-slope form and in slope-intercept form. Example 4: Writing an Equation Using a Table Is the relationship shown by the data linear? If so, model the data with an equation. Use this space to complete pg #5, 8, 9, 11, 16-17, 20-23, 32-35, Page 18

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21 PUTTING IT ALL TOGETHER!! Forms of Linear Equations: Comparing and Contrasting REMEMBER: 1) Lines can be written in many forms. 2) Each form gives specific information about the line. 3) We can convert from form to form. Use the given information to answer each question. a) What is the y-intercept of the line y 7 = 2(x + 1)? b) What is the slope of the graph of 3x 8y = 24? c) What is the x-intercept of y = 2 x + 8? 3 d) When y 2 = 2 (x 4) is written in standard form, what is the coefficient of x? 5 Page 21

22 e) Write the equation of the line in slope-intercept form that has a slope of 4 and passes through 9 the point ( 9, 2). f) Write the equation of the line in slope-intercept form that passes through the points ( 1, 2) and (0, 6). g) Write the equation of the vertical line that passes through the point ( 5, 7). h) Write the equation of the horizontal line that has the same y-intercept as the line y = 7x 10. i) Do the following two lines have the same y-intercept? y + 9 = 3(x + 2) and 4x + y = 3 Page 22

23 Graph each line. a) y = 3 x 8 b) 2x y = 5 4 c) y = 4 d) y + 9 = (x 8) Page 23

24 e) y = 1 (x + 8) f) x = 5 2 g) 4x 3y = 0 h) Does the line y 5 = 2(x 1) pass through each point listed below? Explain. (4,11) (0,1) (5,1) Page 24

Section 6.6 ~ Parallel and Perpendicular Lines Parallel Lines: Slopes of Parallel Lines: Nonvertical lines are parallel if they have slope and y-intercepts. Any two vertical lines are parallel.

25 Section 6.6 ~ Parallel and Perpendicular Lines Parallel Lines: Slopes of Parallel Lines: Nonvertical lines are parallel if they have slope and y-intercepts. Any two vertical lines are parallel. Example: Example 1: Determining Whether Lines Are Parallel Are the graphs of 1 y x 5 and 2x 6y 12 parallel? Explain. 3 Example 2: Writing Equations of Parallel Lines Write an equation for the line that contains (5, 1) and is parallel to 3 y x 4. 5 Page 25

26 Perpendicular Lines: Slopes of Perpendicular Lines: Two lines are perpendicular if their slopes are. - This also means: A vertical and horizontal line are perpendicular. Example: Example 3: Writing Equations for Perpendicular Lines Write an equation for the line that contains (0, -2) and is perpendicular to y 5x 3. Page 26

27 TRY TO APPLY WHAT YOU VE DISCOVERED Are the graphs of the following equations parallel, perpendicular or neither? 1) 6x + 8y = 24 and y = 3 4 x 7 2) y = 1 x 1 and 4x y = 2 4 3) 3x 2y = 18 and 4x + 6y = 0 Write an equation in point-slope form for the line that is parallel/perpendicular to the given line and passes through the given point. 4) parallel to 3x + 5y = 15 through ( 1, 2) 5) perpendicular to y = 4 x + 24 through ( 5, 0) 5 Page 27

28 Write each equation in slope-intercept form. SHOW ALL WORK! 6) Write the equation of the line that has a y-intercept of -2 and is parallel to the graph of y = 1 x ) Write the equation of the line that passes through (5, 6) and is perpendicular to the graph of y = 4x 9. 8) Write the equation of the line that passes through (-4, 1) and is parallel to the graph of 6x + 2y = 4. 9) Write the equation of the line that has a y-intercept of 10 and is perpendicular to the graph of 3x 9y = 15. Page 28

29 Section 1.5 ~ Scatter Plots Scatter plot: Example 1: Making a Scatter Plot The table at the left shows data students collected on their test scores and the number of hours they watched television the previous day. Make a scatter plot for the data. Possible relationships two sets of data may have: Trend line: Examples: Page 29

30 Example 2: Real-World Connections The scatter plot shows the age and asking price of several used mid-sized cars. What type of relationship does the scatter plot show? Page 30

Section 6.7 ~ Scatter Plots and Equations of Lines Example 1: Trend Line Make a scatter plot of the data below. Draw a trend line and write its equation.

Line of Best Fit: Correlation Coefficient: Example 2: Line of Best Fit Use a graphing calculator to find the equation of the line of best fit for the data

31 Section 6.7 ~ Scatter Plots and Equations of Lines Example 1: Trend Line Make a scatter plot of the data below. Draw a trend line and write its equation. Use the equation to predict the wingspan of a hawk that is 28 in. long. Line of Best Fit: Correlation Coefficient: Example 2: Line of Best Fit Use a graphing calculator to find the equation of the line of best fit for the data below. What is the correlation coefficient to three decimal places? Step 1: Use the EDIT feature of the screen. Let 93 correspond to 1993 and 100 correspond to Step 2: Use the CALC feature in the choose LinReg(ax+b). screen and Page 31

32 Use the following data set to create a scatter plot as well as answer the below questions. Average inches of rain per month Number of Umbrellas sold at CVS What type of correlation is there between the two variables? 2. Describe the relationship between the two variables. 3. Predict the weight of a woman who exercises 4 hrs per week. 4. Predict the weight of a woman who exercises 10 hrs per week. 5. Write the equation of a reasonable trend line. Use your graphing calculator to verify. 6. Determine how close your predictions in #3 and #4 were as compared to the value your trend line shows. Page 32

33 Section 7.5 ~ Linear Inequalities Review of Linear Equations - Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-coordinate of the y- intercept of the line. - Standard Form: Ax + By = C, where A,B,C are integers and A is positive. You can easily determine the x-intercept ( C, 0) and the y-intercept A (C, 0) of the line. B - Point-Slope Form: y y 1 = m(x x 1 ) where m is the slope and (x 1, y 1 ) is a point on the line. Review of Solving Inequalities Solve each inequality by isolating the variable. 1) x + 7 > 10 2) 5y 25 3) 4 p < 8p + 4 4) 6m 3(m + 12) Use your solving equations skills to isolate y in each inequality. 5) 3x + y < 5 6) 4y 4(x + 1) 7) 6x 2y 10 8) 10x + 3y > 8 Page 33

34 Something to remember: When graphing inequalities, the line graphed is only a BOUNDARY LINE. Graph each inequality. y 2x 7 y < 1 2 x + 4 NOTES TO SELF: - >, < : -, : - >, : - <, : You Try! Graph each inequality. y 2x + 7 y > 1 2 x + 1 Page 34

the formula for the inequality that is graphed here?

35 What does it mean to be a solution to a linear inequality? Tell if each of the points is a solution to the linear inequality graphed to the left (3,1) (0, 3) (2,0) (1, 4) What is the formula for the inequality that is graphed here? Tell if each of the points is a solution to the linear inequality graphed to the left (4,1) ( 1,5) (2,1) (5,0) What is the formula for the inequality that is graphed here? SUMMARIZE FOR YOURSELF: An ordered pair is a solution to a linear inequality if Page 35

36 *Linear Inequalities will not always be written in slope-intercept form* THINK: How could we graph an inequality that is written in another form?? Graph each inequality. 3x + y 9 2x + 4y < 4 x y 7 6x 2y < 8 Page 36

37 REAL WORLD CONNECTIONS Examples: Suppose your budget for a party allows you to spend no more than $12 on peanuts and cashews. Peanuts cost $2/lb and cashews cost $4/lb. a) Find 3 possible combinations of peanuts and cashews you can buy. b) Write an equation for the situation. c) Graph the relationship. d) Use your 3 combinations from part a to locate points on the graph. What do you notice about the location of those 3 points? e) What is an example of a combination that would NOT work for this situation? Where is that point located on the graph? Page 37

38 You Try! Suppose you spend no more than $24 on meat for a cookout. At your local grocery store, hamburger costs $3.00/lb and chicken wings cost $2/lb. a) Find 3 possible combinations of hamburger and chicken wings you can buy. b) Write an equation for the situation. c) Graph the relationship. THINK Would these phrases be represented by >, <,, or??? No more than = No less than = More than = Less than = Without exceeding budget = Page 38

Algebra 1B. Chapter 6: Linear Equations & Their Graphs Sections 6-1 through 6-7 & 7-5. COLYER Fall Name: Period:

Algebra 1B. Chapter 6: Linear Equations & Their Graphs Sections 6-1 through 6-7 & 7-5. COLYER Fall Name: Period: Chapter 6: Linear Equations & Their Graphs Sections 6-1 through 6-7 & 7-5 COLYER Fall 2016 Name: Period: What s the Big Idea? Analyzing Linear Equations & Inequalities What can I expect to understand when