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Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Lesson 2-1 Relations and Functions Lesson 2-2 Linear Equations Lesson 2-3 Slope Lesson 2-4 Writing Linear Equations Lesson 2-5 Modeling Real-World Data: Using Scatter Plots Lesson 2-6 Special Functions Lesson 2-7 Graphing Inequalities Example 1 Domain and Range Example 2 Vertical Line Test Example 3 Graph Is a Line Example 4 Graph Is a Curve Example 5 Evaluate a Function State the domain and range of the relation shown in the graph. Is the relation a function? The relation is {(1, 2), (3, 3), (0, –2), (–4, 0), (–3, 1)}. Answer: The domain is {–4, –3, 0, 1, 3}. The range is {–2, 0, 1, 2, 3}. Each member of the domain is paired with exactly one member of the range, so this relation is a function. State the domain and range of the relation shown in the graph. Is the relation a function? Answer: The domain is {–3, 0, 2, 3}. The range is {–2, –1, 0, 1}. Yes, the relation is a function. Transportation The table shows the average fuel efficiency in miles per gallon for light trucks for several years. Graph this information and determine whether it represents a function. Year Fuel Efficiency (mi/gal) 1995 20.5 1996 1997 1998 1999 20.8 20.6 20.9 20.5 2000 2001 20.5 20.4 Year Fuel Efficiency (mi/gal) 1995 20.5 1996 1997 1998 1999 20.8 20.6 20.9 20.5 2000 2001 20.5 20.4 Use the vertical line test. Notice that no vertical line can be drawn that contains more than one of the data points. Answer: Yes, this relation is a function. Health The table shows the average weight of a baby for several months during the first year. Graph this information and determine whether it represents a function. Age (months) Weight (pounds) 1 12.5 2 4 6 9 16 22 24 25 12 26 Answer: Yes, this relation is a function. Graph the relation represented by Make a table of values to find ordered pairs that satisfy the equation. Choose values for x and find the corresponding values for y. Then graph the ordered pairs. x y –1 0 1 2 –4 –1 2 5 (2, 5) (1, 2) (0, –1) (–1, –4) Find the domain and range. Since x can be any real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the line shown. Notice that every real number is the x-coordinate of some point on the line. Also, every real number is the y-coordinate of some point on the line. (2, 5) (1, 2) (0, –1) (–1, –4) Answer: The domain and range are both all real numbers. Determine whether the relation is a function. This graph passes the vertical line test. For each x value, there is exactly one y value. (2, 5) (1, 2) Answer: Yes, the equation represents a function. (0, –1) (–1, –4) a. Graph Answer: b. Find the domain and range. Answer: The domain and range are both all real numbers. c. Determine whether the relation is a function. Answer: Yes, the equation is a function. Graph the relation represented by Make a table. In this case, it is easier to choose y values and then find the corresponding values for x. Then sketch the graph, connecting the points with a smooth curve. x y 5 2 1 2 5 –2 –1 0 1 2 (5, 2) (2, 1) (1, 0) (2, –1) (5, –2) Find the domain and range. Every real number is the y-coordinate of some point on the graph, so the range is all real numbers. But, only real numbers that are greater than or equal to 1 are x-coordinates of points on the graph. Answer: The domain is The range is all real numbers. (5, 2) (2, 1) (1, 0) (2, –1) (5, –2) . Determine whether the relation is a function. x y 5 2 1 2 5 –2 –1 0 1 2 (5, 2) (2, 1) (1, 0) (2, –1) (5, –2) You can see from the table and the vertical line test that there are two y values for each x value except x = 1. Answer: The equation a function. does not represent a. Graph Answer: b. Find the domain and range. Answer: The domain is {x|x –3}. The range is all real numbers. c. Determine whether the relation is a function. Answer: No, the equation represent a function. does not Given , find Original function Substitute. Simplify. Answer: Given find Original function Substitute. Multiply. Simplify. Answer: Given , find Original function Substitute. Answer: Given find each value. and a. Answer: 6 b. Answer: 0.625 c. Answer: Example 1 Identify Linear Functions Example 2 Evaluate a Linear Function Example 3 Standard Form Example 4 Use Intercepts to Graph a Line State whether Explain. is a linear function. Answer: This is a linear function because it is in the form State whether Explain. is a linear function. Answer: This is not a linear function because x has an exponent other than 1. State whether Explain. is a linear function. Answer: This is a linear function because it can be written as State whether each function is a linear function. Explain. a. Answer: yes; b. Answer: No; x has an exponent other than 1. c. Answer: No; two variables are multiplied together. Meteorology The linear function can be used to find the number of degrees Fahrenheit, f (C), that are equivalent to a given number of degrees Celsius, C. On the Celsius scale, normal body temperature is 37C. What is normal body temperature in degrees Fahrenheit? Original function Substitute. Simplify. Answer: Normal body temperature, in degrees Fahrenheit, is 98.6F. There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree? Divide 180 Fahrenheit degrees by 100 Celsius degrees. Answer: 1.8F = 1C Meteorology The linear function can be used to find the distance d(s) in miles from a storm, based on the number of seconds s that it takes to hear thunder after seeing lightning. a. If you hear thunder 10 seconds after seeing lightning, how far away is the storm? Answer: 2 miles b. If the storm is 3 miles away, how long will it take to hear thunder after seeing lightning? Answer: 15 seconds Write in standard form. Identify A, B, and C. Original equation Subtract 3x from each side. Multiply each side by –1 so that A 0. Answer: and Write and C. in standard form. Identify A, B, Original equation Subtract 2y from each side. Multiply each side by –3 so that the coefficients are all integers. Answer: and Write and C. in standard form. Identify A, B, Original equation Subtract 4 from each side. Divide each side by 2 so that the coefficients have a GCF of 1. Answer: and Write each equation in standard form. Identify A, B, and C. a. Answer: and b. Answer: and c. Answer: and Find the x-intercept and the y-intercept of the graph of Then graph the equation. The x-intercept is the value of x when Original equation Substitute 0 for y. Add 4 to each side. Divide each side by –2. The x-intercept is –2. The graph crosses the x-axis at (–2, 0). Likewise, the y-intercept is the value of y when Original equation Substitute 0 for x. Add 4 to each side. The y-intercept is 4. The graph crosses the y-axis at (0, 4). Use the ordered pairs to graph this equation. Answer: The x-intercept is –2, and the y-intercept is 4. (0, 4) (–2, 0) Find the x-intercept and the y-intercept of the graph of Then graph the equation. Answer: The x-intercept is –2, and the y-intercept is 6. Example 1 Find Slope Example 2 Use Slope to Graph a Line Example 3 Rate of Change Example 4 Parallel Lines Example 5 Perpendicular Line Find the slope of the line that passes through (1, 3) and (–2, –3). Then graph the line. Slope formula and Simplify. Graph the two ordered pairs and draw the line. Use the slope to check your graph by selecting any point on the line. Then go up 2 units and right 1 unit or go down 2 units and left 1 unit. This point should also be on the line. Answer: The slope of the line is 2. (1, 3) (–2, –3) Find the slope of the line that passes through (2, 3) and (–1, 5). Then graph the line. Answer: The slope of the line is Graph the line passing through (1, –3) with a slope of Graph the ordered pair (1, –3). Then, according to the slope, go down 3 units and right 4 units. Plot the new point at (5, –6). Draw the line containing the points. (1, –3) (5, –6) Graph the line passing through (2, 5) with a slope of –3. Answer: Communication Refer to the graph. Find the rate of change of the number of radio stations on the air in the United States from 1990 to 1998. Slope formula Substitute. Simplify. Answer: Between 1990 and 1998, the number of radio stations on the air in the United States increased at an average rate of 0.225(1000) or 225 stations per year. Computers Refer to the graph. Find the rate of change of the number of households with computers in the United States from 1984 to 1998. Answer: The rate of change is 2.9 million households per year. Graph the line through (1, –2) that is parallel to the line with the equation The x-intercept is –2 and the y-intercept is 2. Use the intercepts to graph The line rises 1 unit for every 1 unit it moves to the right, so the slope is 1. Now, use the slope and the point at (1, –2) to graph the line parallel to (2, –1) (1, –2) Graph the line through (2, 3) that is parallel to the line with the equation Answer: Graph the line through (2, 1) that is perpendicular to the line with the equation The x-intercept is or 1.5 and the y-intercept is –1. Use the intercepts to graph 2x – 3y = 3 The line rises 1 unit for every 1.5 units it moves to the right, so the slope is or Graph the line through (2, 1) that is perpendicular to the line with the equation The slope of the line perpendicular is the opposite reciprocal of or Start at (2, 1) and go down 3 units and right 2 units. Use this point and (2, 1) to graph the line. (2, 1) 2x – 3y = 3 (4, –2) Graph the line through (–3, 1) that is perpendicular to the line with the equation Answer: Example 1 Write an Equation Given Slope and a Point Example 2 Write an Equation Given Two Points Example 3 Write an Equation for a Real-World Situation Example 4 Write an Equation of a Perpendicular Line Write an equation in slope-intercept form for the line that has a slope of and passes through (5, –2). Slope-intercept form Simplify. Add 3 to each side. Answer: The y-intercept is 1. So, the equation in slope-intercept form is Write an equation in slope-intercept form for the line that has a slope of Answer: and passes through (–3, –1). Multiple-Choice Test Item What is an equation of the line through (2, –3) and (–3, 7)? A B C D Read the Test Item You are given the coordinates of two points on the line. Notice that the answer choices are in slope-intercept form. Solve the Test Item First, find the slope of the line. Slope formula Simplify. The slope is –2. That eliminates choices B and C. Then use the point-slope formula to find an equation. Point-slope form you can use either point for . Distributive Property Subtract 3 from each side. Answer: D Multiple-Choice Test Item What is an equation of the line through (2, 5) and (–1, 3)? A C B D Answer: C Sales As a part-time salesperson, Jean Stock is paid a daily salary plus commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. Write a linear equation to model this situation. Let x be her sales and let y be the amount of money she makes. Use the points (100, 58) and (300, 78) to make a graph to represent the situation. Slope formula Simplify. Now use the slope and either of the given points with the point-slope form to write the equation. Point-slope form Distributive Property Add 58 to each side. Answer: The slope-intercept form of the equation is What are Ms. Stock’s daily salary and commission rate? The y-intercept of the line is 48. The y-intercept represents the money Jean would make if she had no sales. Thus, $48 is her daily salary. The slope of the line is 0.1. Since the slope is the coefficient of x, which is her sales, she makes 10% commission. Answer: Ms. Stock’s daily salary is $48, and she makes a 10% commission. How much would Jean make in a day if her sales were $500? Find the value of y when Use the equation you found in Example 3a. Replace x with 500. Simplify. Answer: She would make $98 if her sales were $500. Sales The student council is selling coupon books to raise money for the Humane Society. If the group sells 200 books, they will receive $150 dollars. If they sell 500 books, they will make $375. a. Write a linear equation to model this situation. Answer: b. Find the percentage of the proceeds that the student council receives. Answer: 75% c. If they sold 1000 books, how much money would they receive to donate to the Humane Society? Answer: $750 Write an equation for the line that passes through (3, –2) and is perpendicular to the line whose equation is The slope of the given line is –5. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular line is Use the point-slope form and the ordered pair (3, –2) to write the equation. Point-slope form Distributive Property Subtract 2 from each side. Answer: An equation of the line is Write an equation for the line that passes through (3, 5) and is perpendicular to the line whose equation is Answer: Example 1 Draw a Scatter Plot Example 2 Find and Use a Prediction Equation Education The table below shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data. Years Since 1985 0 3 6 9 12 15 Percent 20 18 15 15 14 13 Source: U.S. News & World Report Graph the data as ordered pairs, with the number of years since 1985 on the horizontal axis and the percentage on the vertical axis. Safety The table below shows the approximate percent of drivers who wear seat belts in various years since 1994. Make a scatter plot of the data. Years Since 1994 0 Percent 57 58 61 64 69 68 71 73 1 2 3 4 5 6 7 Source: National Highway Traffic Safety Administration Education The table and scatter plot below show the approximate percent of students who sent applications to two colleges in various years since 1985. Draw a line of fit for the data. How well does the line fit the data? Years Since 1985 0 3 6 9 12 15 Percent 20 18 15 15 14 13 Source: U.S. News & World Report The points (3, 18) and (15, 13) appear to represent the data well. Draw a line through these two points. Education The table and scatter plot below show the approximate percent of students who sent applications to two colleges in various years since 1985. Draw a line of fit for the data. How well does the line fit the data? Years Since 1985 0 3 6 9 12 15 Percent 20 18 15 15 14 13 Source: U.S. News & World Report Answer: Except for (6, 15), this line fits the data fairly well. Find a prediction equation. What do the slope and y-intercept indicate? Find an equation of the line through (3, 18) and (15, 13). Begin by finding the slope. Slope formula Substitute. Simplify. Point-slope form Distributive Property Add 18 to each side. Answer: One prediction equation is The slope indicates that the percent of students sending applications to two colleges is falling at about 0.4% each year. The y-intercept indicates that the percent in 1985 should have been about 19%. Predict the percent in 2010. The year 2010 is 25 years after 1985, so use the prediction equation to find the value of y when Prediction equation Simplify. Answer: The model predicts that the percent in 2010 should be about 9%. How accurate is this prediction? Answer: The fit is only approximate, so the prediction may not be very accurate. Safety The table and scatter plot show the approximate percent of drivers who wear seat belts in various years since 1994. Years Since 1994 0 Percent 57 58 61 64 69 68 71 73 1 2 3 4 5 6 7 Source: National Highway Traffic Safety Administration a. Draw a line of fit for the data. How well does the line fit the data? Answer: Except for (4, 69), this line fits the data very well. b. Find a prediction equation. What do the slope and y-intercept indicate? Answer: Using (1, 58) and (7, 73), an equation is y = 2.5x + 55.5. The slope indicates that the percent of drivers wearing seatbelts is increasing at a rate of 2.5% each year. The y-intercept indicates that, according to the trend of the rest of the data, the percent of drivers who wore seatbelts in 1994 was about 56%. c. Predict the percent of drivers who will be wearing seat belts in 2005. Answer: 83% d. How accurate is the prediction? Answer: Except for the outlier, the line fits the data very well, so the predicted value should be fairly accurate. Example 1 Step Function Example 2 Constant Function Example 3 Absolute Value Functions Example 4 Piecewise Function Example 5 Identify Functions Psychology One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this solution. Explore The total charge must be a multiple of $85, so the graph will be the graph of a step function. Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on. Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph. x C(x) 85 170 255 340 425 Answer: Examine Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint. Sales The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation. Answer: Graph For every value of horizontal line. g(x) = –3 x g(x) –2 –3 0 1 0.5 –3 –3 –3 The graph is a Answer: Graph Answer: Graph and on the same coordinate plane. Determine the similarities and differences in the two graphs. Find several ordered pairs for each function. x |x–3| x |x+2| 0 3 –4 2 1 2 –3 1 2 1 –2 0 3 0 –1 1 4 1 0 2 5 2 1 3 Graph the points and connect them. Answer: • The domain of both graphs is all real numbers. • The range of both graphs is • The graphs have the same shape, but different x-intercepts. • The graph of g(x) is the graph of f(x) translated left 5 units. Graph and on the same coordinate plane. Determine the similarities and differences in the two graphs. Answer: • The domain of both graphs is all real numbers. • The range of is The range of is • The graphs have the same shape, but different y-intercepts. • The graph of g(x) is the graph of f(x) translated up 5 units. Graph and range. Identify the domain Step 1 Graph the linear function for Since 3 satisfies this inequality, begin with a closed circle at (3, 2). Graph and range. Identify the domain Step 2 Graph the constant function Since x does not satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right. Graph and range. Identify the domain Answer: The function is defined for all values of x, so the domain is all real numbers. The values that are y-coordinates of points on the graph are all real numbers less than or equal to –2, so the range is Graph and range. Answer: The domain is all real numbers. The range is Identify the domain Determine whether the graph represents a step function, a constant function, an absolute value function, or a piecewise function. Answer: Since this graph consists of different rays and segments, it is a piecewise function. Determine whether the graph represents a step function, a constant function, an absolute value function, or a piecewise function. Answer: Since this graph is V-shaped, it is an absolute value function. Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function. a. b. Answer: constant function Answer: absolute value function Example 1 Dashed Boundary Example 2 Solid Boundary Example 3 Absolute Value Inequality Graph The boundary is the graph of Since the inequality symbol is <, the boundary will be dashed. Use the slope-intercept form, Graph Test (0, 0). Original inequality true Shade the region that contains (0, 0). Graph Answer: Education The SAT has two parts. One tutoring company advertises that it specializes in helping students who have a combined score on the SAT that is 900 or less. Write an inequality to describe the combined scores of students who are prospective tutoring clients. Let x be the first part of the SAT and let y be the second part. Since the scores must be 900 or less, use the symbol. The 1st part x Answer: and 2nd part together are less than or equal to 900. y 900 Graph the inequality. Since the inequality symbol is , the graph of the related linear equation is solid. This is the boundary of the inequality. Graph the inequality. Test (0, 0). Original inequality true Graph the inequality. Shade the region that contains (0, 0). Since the variables cannot be negative, shade only the part in the first quadrant. Does a student with a verbal score of 480 and a math score of 410 fit the tutoring company’s guidelines? The point (480, 410) is in the shaded region, so it satisfies the inequality. Answer: Yes, this student fits the tutoring company’s guidelines. Class Trip Two social studies classes are going on a field trip. The teachers have asked for parent volunteers to also go on the trip as chaperones. However, there is only enough seating for 60 people on the bus. a. Write an inequality to describe the number of students and chaperones that can ride on the bus. Answer: b. Graph the inequality. Answer: c. Can 45 students and 10 chaperones go on the trip? Answer: yes Graph Since the inequality symbol is , the graph of the related equation is solid. Graph the equation. Test (0, 0). Original inequality true Shade the region that contains (0, 0). Graph Answer: Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Algebra 2 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.algebra2.com/extra_examples. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. 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