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 Chapter 6 is complete? Linear equations and inequalities represent the relationship between two variables that have a constant rate of change. What questions will I be able to answer when Chapter 6 is complete? 1) What is the slope of a line? 2) How do you find slope of a line given the coordinates of two of its points? 3) How does one write linear equations in point-slope form? 4) How does one draw a scatter plot and find the equation of the best-fit line for data? 5) How does one solve problems using linear models? 6) How does one graph linear equations? 7) How does one use slope to determine if two lines are parallel or perpendicular? 8) How does one graph linear inequalities? 9) How can one be sure that an ordered pair represents a solution to a linear equation or a linear inequality? Page 2
6-1: Rate of Change and Slope Rate of Change: NOTE: Example: For the data in the table below, is the rate of change for each pair of consecutive days the same? What does the rate of change represent? Graph the ordered pairs from the table on the coordinate plane to the right. If we were given just the graph, how could we find the rate of change? Remember: In a graph, the dependent variable is on the and the independent variable is on the. Page 3
Example: The graph shows the altitude of an airplane as it comes in for a landing. Find the rate of change. Explain what it means. You Try!! Find each rate of change. Explain what the rate means. 1) 2) Weight Price of (lb) Apples 0 $1.50 1 $3.00 2 $4.50 3 $6.00 3) 4) Page 4
EXPLORING RATE OF CHANGE The diagram to the right shows the side view of a ski lift. 1) What is the vertical change from A to B? From B to C? From C to D? 2) What is the horizontal change from A to B? From B to C? From C to D? 3) Find the ratio of the vertical change to the horizontal change for each section of the ski lift. 4) Which section is steepest? Explain. Finding Slope Slope of a line: Two ways to determine the slope of a line: 1) 2) Page 5
Finding Slope Using A Graph Examples: Find the slope using rise run. FORMULA: Find the slope using the formula. Find the slope using whichever method you choose. Page 6
Four different types of slope: PRACTICE: Find the slope of each line. Page 7
Find Slope Given Two Points FORMULA: Examples: Find the slope of the line between the two points. a) (2, 5), ( 4, 7) b) ( 1, 4), (3, 2) c) ( 8, 6), (4, 6) d) ( 1, 5), ( 1, 8) You Try! 1) ( 2, 9), (4, 2) 2) (1, 8), (1, 2) 3) (8, 10), ( 4, 6) 4) ( 8, 4), ( 8, 8) Page 8
6-2: Slope-Intercept Form Slope of a line: Y-Intercept of a line: SLOPE-INTERCEPT FORM: Identifying Slope and Y-Intercept a) y = 2x + 8 b) y = 2 3 x 5 Slope: Y-Intercept: Slope: Y-Intercept: Writing an Equation in Slope-Intercept Form a) Write the equation of the line with a slope of 1 2 and a y-intercept of (0, 3). b) Write the equation of a line with a slope of 1 and a y-intercept of (0, 0). Page 9
c) Write the equation of a line that intersects the points (1, 7) and ( 3, 3) with a y-intercept of (0, 5). d) Write the equation of a line that intersects the points ( 2, 6) and (0, 3). YOU TRY!! 1) What is the slope and y-intercept of the line with the equation y = x 8? 2) Write the equation of the line with a slope of 1 and a y-intercept of (0, 10). 3) Write the equation of the line that intersects the points ( 1, 4) and ( 3, 2) with a y-intercept of (0, 24). 4) Write the equation of the line that intersects the points ( 10, 4) and (0, 2). Page 10
Writing an Equation from a Graph Write the equation of each line in slope-intercept form. a) b) STEPS TO WRITING EQUATIONS IN SLOPE-INTERCEPT FORM 1) 2) 3) Graphing Lines with Equations in Slope-Intercept Form 1) y = 4 x 3 b) y = 2x 5 Page 11
STEPS TO GRAPHING EQUATIONS IN SLOPE-INTERCEPT FORM 1) 2) 3) YOU TRY!! Write the equation of each line in slope-intercept form. 1) 2) Graph each equation. 3) y = x + 8 4) y = 5 2 x 7 Page 12
SLOPE-INTERCEPT FORM: Find the slope and y-intercept of each equation. a) y 2 = 3x b) 2y = 6(5 3x) Use the slope and y-intercept to graph each equation. c) y = 7 3x d) 2y + 2x = 0 Page 13
YOU TRY!! Find the slope and y-intercept of each equation. 1) 2y 6 = 8x 2) 2y = 18x + 1 Use the slope and y-intercept to graph each equation. 3) y = 3x + 9 4) 4x 8y = 8 5) 9y = 3x 6) 10y = 25 + 6x Page 14
6-3 REAL WORLD CONNECTIONS Examples: a. While traveling on I-95, you set your cruise control to 60 mph. Graph the relationship between the time spent driving and the distance covered. What is the equation that models this situation? Equation: b. The base pay of a water-delivery person is $210 per week. He also earns a 20% commission on any sale he makes. Graph this relationship. What equation models this situation? Equation: Page 15
6-4: Standard Form **REMEMBER: The equation of a line can be written in many different forms. STANDARD FORM OF A LINE: *In different situations, it is useful to transform equations from one form to another. Transform each equation to Standard Form. a) y = 3x + 10 b) y = 3 4 x + 2 c) y = 2 5 x d) y = 7 5 x 1 2 Page 16
You Try!! 1) y = x + 1 2) y = 4(x 4) 3) y = 1 5 x 3 4) y = 2 5 x 1 3 5) y = 9x 6) y = 1 3 (x + 2 3 ) Page 17
Y-Intercept of a line: X-Intercept of a line: Finding X- and Y-Intercepts a) 2x 4y = 8 b) 2x + 4y = 8 X-Intercept: X-Intercept: Y-Intercept: Y-Intercept: c) x 2y = 6 d) 6x y = 3 X-Intercept: X-Intercept: Y-Intercept: Y-Intercept: Page 18
Graphing Lines Using Intercepts How many points do you need to be able to draw a line? a) 2x + 3y = 12 b) 3x 5y = 30 Graphing Horizontal and Vertical Lines b) y = 3 e) x = 2 Page 19
YOUR TURN!! Find the x- and y-intercepts. 1) 5x + 2y = 10 2) 4x 9y = 12 X-Intercept: X-Intercept: Y-Intercept: Y-Intercept: Graph each line from standard form. 3) 3x y = 3 4) 8x + 6x = 12 5) y = 6 6) x = 7 Page 20
6-5: Point-Slope Form POINT-SLOPE FORM OF A LINE: Where does this formula come from?? Graphing Using Point-Slope Form c) y 5 = 1 (x 4) b) y + 8 = 2(x + 1) 2 Page 21
You Try!!! 1) y 3 = 3(x + 4) 2) y + 6 = 2 (x 4) 5 3) y + 1 = (x + 9) 4) y 3 = 1 2 (x) Page 22
Writing an Equation in Point-Slope Form a) Write an equation of the line with slope -3 that passes through the point (-1, 7). b) Write an equation of the line that passes through the points (-3, -4) and (9, 0). c) Write an equation of the line that passes through the points (2, 3) and (-1, -5). d) Write an equation in point-slope form of the line depicted in the graph below. Then convert to slope-intercept form. Page 23
e) Write an equation in point-slope form of the line depicted in the graph below. Then convert to slope-intercept form. f) Write an equation of the horizontal line that passes through the point (9, 8). g) Write an equation of the vertical line that passes through the point (2, 3). Page 24
PRACTICE GRAPHING ALL TYPES OF EQUATIONS Graph each equation based on the form it is written in. 1) y = 2x + 8 2) x 2y = 2 HINT:: Slope-Intercept Form HINT:: Standard Form 3) y 6 = 2 (x + 8) 4) y + 4 = (x) 3 HINT:: Point-Slope Form HINT:: Point-Slope Form Page 25
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 5 of x? Page 26
e) Write the equation of the line in slope-intercept form that has a slope of 4 9 and passes through 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 27
Graph each line. a) y = 3 x 8 b) 2x y = 5 4 c) y = 4 d) y + 9 = (x 8) Page 28
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 29
6-7: Scatter Plots and Equations of Lines Positive Correlation: Negative Correlation: No Correlation: Use the following data set to create a scatter plot. Label each axis and give the plot a title. Average inches of rain per month Number of Umbrellas sold at CVS 8 5.5 6 5 2.5 3.5 4 9 7 8.5 15 12 14 13 6 9 5 16 14 15 a) What type of correlation is there between the two variables? b) Describe the relationship between the two variables. c) Predict the number of umbrellas sold if there was an average of 3 inches of rain. d) Predict the number of umbrellas sold if there was an average of 10 inches of rain. *Check yourself using a graphing calculator!* Page 30
Trend Line: Let s draw the trend line for the scatter plot on the previous page What equation would represent the trend line we drew for the scatter plot on the front? Practice! Find an equation of a reasonable trend line for each scatter plot. 1) Make a scatter plot for the data below. 2) Draw a trend line. 3) Write its equation. Length (in.) Wingspan (in.) 21 36 21 41 18 38 24 46 16 31 19 39 17 35 19 46 Page 31
Use the following data set to create a scatter plot as well as answer the below questions. Hours of exercise per week 1 3 6 5 2.5 3.5 8 9 7 8.5 Weight 165 150 142 140 158 145 135 138 140 127 1. 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. 6. Determine how close your predictions in #3 and #4 were as compared to the value your trend line shows. Page 32
6-6: Parallel and Perpendicular Lines Graph the following pairs of equations on the same plane. y = 3x + 1 y = 2x + 5 y = 3x 8 y = 1 x 2 2 What do you notice??? What do you notice??? y = 2 x + 7 3 y = x + 2 y = 3 x 2 y = x 2 What do you notice??? What do you notice??? Page 33
Definitions and Important Patterns. Parallel Lines: Lines that do not intersect. Find the slope of each line and compare. What pattern can you determine about Slopes of Parallel Lines?? Perpendicular Lines: Lines that intersect at a 90 degree angle. Find the slope of each line and compare. What pattern can you determine about Slopes of Perpendicular Lines?? *If you are not finding a pattern. Look it up!!* Here is a great video that may help you understand: http://www.youtube.com/watch?v=rew54k6myuo Page 34
TRY TO APPLY WHAT YOU VE DISCOVERED Compare the slope of each equation. 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 35
Parallel and Perpendicular Lines Practice Problems Write each equation in slope-intercept form. SHOW ALL WORK! 1) Write the equation of the line that has a y-intercept of -2 and is parallel to the graph of y = 1 x + 6. 2 2) Write the equation of the line that passes through (5, 6) and is perpendicular to the graph of y = 4x 9. 3) Write the equation of the line that passes through (-4, 1) and is parallel to the graph of 6x + 2y = 4. 4) Write the equation of the line that has a y-intercept of 10 and is perpendicular to the graph of 3x 9y = 15. Page 36
Determine if each pair of lines is parallel, perpendicular, or neither. Explain your answer. 5) y = 3x + 2 6) 4x + 8y = 1 9x + 3y = 6 y = 1 x 1. 2 7) y = 2 8) 8x 2y = 2 x = 9 4x y = 10 Page 37
NOTES: 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 38
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 + 4 Page 39
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 40
*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 41
PRACTICE! PRACTICE! PRACTICE! Page 42
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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 45
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 46