Notes for Algebra1 Chapter 4

Graphing Equations in Slope-intercept form 4-1 notes for Algebra 1 Graphing Equations in Slope-intercept form

Slope-intercept form 𝑦=𝑚𝑥+𝑏 m – the slope of the line b – the y-intercept of the line (where the line crosses the y-axis) (x, y) all the points that lie on the line.

Example 1 pg. 216 Write and graph an equation Write an equation in slope-intercept form of the line with a slope of 1 4 and a y-intercept of −1. Then graph the equation.

Example 1 pg. 216 Write and graph an equation Write an equation in slope-intercept form of the line with a slope of 1 4 and a y-intercept of −1. Then graph the equation. 𝑦= 1 4 𝑥−1 3 2 1 4 -1

Example 2 pg. 217 Graph Linear Equations

Example 2 pg. 217 Graph Linear Equations -2 -1 3 4 5 6 7 -3 -4 -5

Constant Functions 𝑦=0𝑥+𝑏 or 𝑦=𝑏 These are horizontal lines and will not cross the x-axis.

Example 3 pg. 217 Graph Linear Equations

Example 3 pg. 217 Graph Linear Equations -4 -3 -2 -1 2 3 4 5 -5 -6 Graph 𝑦=−7

Example 4 pg. 218 Write an Equation in Slope-Intercept Form 3 2 1 -4 -3 -2 -1 5 A.) 𝑦= 1 2 𝑥+3 B.) 𝑦=2𝑥−3 C.) 𝑦= 1 2 𝑥−3 D.) 𝑦=−2𝑥−3

Example 4 pg. 218 Write an Equation in Slope-Intercept Form 3 2 1 -4 -3 -2 -1 5 A.) 𝑦= 1 2 𝑥+3 B.) 𝑦=2𝑥−3 C.) 𝑦= 1 2 𝑥−3 D.) 𝑦=−2𝑥−3

Example 5 pg. 219 Write and Graph a Linear Equation HEALTH The ideal maximum heart rate for a 25-year-old exercising to burn fat is 117 beats per minute. For every five years older than 25, that ideal rate drops three beats per minute. 1.) Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat. 2.) Graph the equation. 3.) Find the ideal maximum heart rate for a 55-year-old person exercising to burn fat.

Example 5 pg. 219 Write and Graph a Linear Equation HEALTH The ideal maximum heart rate for a 25-year-old exercising to burn fat is 117 beats per minute. For every five years older than 25, that ideal rate drops three beats per minute. 1.) Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat. 𝑅=− 3 5 𝑎+117, where R is the ideal maximum heart rate for a 25-year-old and a is the number of years older than 25.

Example 5 pg. 219 Write and Graph a Linear Equation 2.) Graph the equation. Maximum Heart Rate R 117 116 115 114 113 112 111 1 2 3 4 5 6 7 8 a Years older than 25

Example 5 pg. 219 Write and Graph a Linear Equation HEALTH The ideal maximum heart rate for a 25-year-old exercising to burn fat is 117 beats per minute. For every five years older than 25, that ideal rate drops three beats per minute. 3.) Find the ideal maximum heart rate for a 55-year-old person exercising to burn fat. 99 beats per minute

4-1 pg. 220 17-49o, 50, 51, 53-57o, 69-81(x3)

Writing Equations in Slope-Intercept Form 4-2 Notes for Algebra 1 Writing Equations in Slope-Intercept Form

Example 1 Write an equation given the slope and a point. Write an equation of a line that passes through 2, −3 with a slope of 1 2 .

Example 1 Write an equation given the slope and a point. Write an equation of a line that passes through 2, −3 with a slope of 1 2 . 𝑦= 1 2 𝑥−4

Example 2 pg. 227 Write an equation given two points 1.) −3, −4 and −2, −8 2.) 6, −2 and 3, 4

Example 2 pg. 227 Write an equation given two points 1.) −3, −4 and −2, −8 2.) 6, −2 and 3, 4 𝑦=−4𝑥−16 𝑦=−2𝑥+10

Constraint A condition that a solution must satisfy.

Example 3 pg. 228 Use slope-intercept form ECONOMY During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July. Write a linear equation to predict Malik’s cost of gasoline the first of any month during the year, using 1 to represent January.

Example 3 pg. 228 Use slope-intercept form ECONOMY During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July. Write a linear equation to predict Malik’s cost of gasoline the first of any month during the year, using 1 to represent January. 𝑦=0.22𝑥+1.88

Linear extrapolation The process that uses a linear equation to make predictions about values that are beyond the range of data.

Example 4 pg. 228 Predict from Slope- Intercept form. ECONOMY On average, Malik uses 25 gallons of gasoline per month. He budgeted $100 for gasoline October. Use the prediction equation in Example 3 to determine if Malik will have to add to his budget. Explain.

Example 4 pg. 228 Predict from Slope- Intercept form. ECONOMY On average, Malik uses 25 gallons of gasoline per month. He budgeted $100 for gasoline October. Use the prediction equation in Example 3 to determine if Malik will have to add to his budget. Explain. If gasoline prices increase at the same rate, a gallon will cost $4.08 in October. 25 gallons at this price is $102, so Malik will have to add at least $2 to his budget.

4-2 pg. 229 11-31o, 32, 33-39o, 54-75(x3)

Writing Equations in point-slope form. 4-3 Notes for Algebra 1 Writing Equations in point-slope form.

Point-Slope form 𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1 m– is the slope of the line 𝑥 1 , 𝑦 1 -- is a given point on the line. 𝑥, 𝑦 -- represents all points on the line.

Example 1 pg. 233 Write an Graph an Equation in Point-Slope form. Write an equation in point-slope form for the line that passes through −2, 0 with a slope of − 3 2 . Then graph the equation.

Example 1 pg. 233 Write an Graph an Equation in Point-Slope form. Write an equation in point-slope form for the line that passes through −2, 0 with a slope of − 3 2 . Then graph the equation. 𝑦−0=− 3 2 (𝑥+2) 4 3 2 1 -4 -3 -2 -1 5

Example 2 pg. 234 Standard Form Write 𝑦= 3 4 𝑥−5 in standard form.

Example 2 pg. 234 Standard Form Write 𝑦= 3 4 𝑥−5 in standard form. 3𝑥−4𝑦=20

Example 3 pg. 234 Slope-Intercept Form Write 𝑦−5= 4 3 𝑥−3 in slope-intercept form.

Example 3 pg. 234 Slope-Intercept Form Write 𝑦−5= 4 3 𝑥−3 in slope-intercept form. 𝑦= 4 3 𝑥+1

Example 4 pg. 235 Point-slope form and Standard form. GEOMETRY The figure shows trapezoid ABCD with bases 𝐴𝐵 and 𝐶𝐷 . 1.) Write an equation in point- slope form for the line containing the side 𝐵𝐶 2.) Write an equation in standard form for the same line. 𝐴 −2, 3 𝐵 4, 3 𝐷 1, −2 𝐶 4 6 ,−2

Example 4 pg. 235 Point-slope form and Standard form. GEOMETRY The figure shows trapezoid ABCD with bases 𝐴𝐵 and 𝐶𝐷 . 1.) Write an equation in point- slope form for the line containing the side 𝐵𝐶 𝑦−3=− 5 2 𝑥−4 or 𝑦+2=− 5 2 𝑥−6 2.) Write an equation in standard form for the same line. 5𝑥+2𝑦=26 𝐴 −2, 3 𝐵 4, 3 𝐷 1, −2 𝐶 4 6 ,−2

4-3 pg. 236 11-33o, 34-39, 54-75(x3)

Parallel and Perpendicular lines 4-4 Notes for Algebra 1 Parallel and Perpendicular lines

Parallel Lines Lines in the same plane that do not intersect. Non-vertical Parallel lines have the same slope.

Example 1 pg. 239 Parallel Lines through a given point. Write an equation in slope-intercept form for the line that passes through 4, −2 and is parallel to the graph of 𝑦= 1 2 𝑥−4.

Example 1 pg. 239 Parallel Lines through a given point. Write an equation in slope-intercept form for the line that passes through 4, −2 and is parallel to the graph of 𝑦= 1 2 𝑥−7. 𝑦= 1 2 𝑥−4

Perpendicular lines Lines that intersect at right angles. The slopes of non-vertical perpendicular lines are opposite reciprocals. That means their product is −1.

Example 2 pg. 240 slopes of perpendicular lines T 2 1 P -5 -4 E -2 -1 3 4 5 6 7 R -3 -6 -7 Z -8 -9 A -10 GEOMETRY The height of a trapezoid is the length of a segment that is perpendicular to both bases. In trapezoid ARTP, 𝐴𝑃 and 𝑅𝑇 are bases. 1.) Can 𝐸𝑍 be used to measure the height of the trapezoid? Explain. 2.) Are the bases parallel

Example 2 pg. 240 slopes of perpendicular lines T 2 1 P -5 -4 E -2 -1 3 4 5 6 7 R -3 -6 -7 Z -8 -9 A -10 GEOMETRY The height of a trapezoid is the length of a segment that is perpendicular to both bases. In trapezoid ARTP, 𝐴𝑃 and 𝑅𝑇 are bases. 1.) Can 𝐸𝑍 be used to measure the height of the trapezoid? Explain. No, the slope of 𝑅𝑇 is 1, and the slope of 𝑅𝑇 is −7. since they are not perpendicular, it cannot be used as the height 2.) Are the bases parallel Yes, both have a slope of 1.

Example 3 pg. 241 Parallel or Perpendicular lines Determine whether the graphs of 3𝑥+𝑦=12, 𝑦= 1 3 𝑥+2, and 2𝑥−6𝑦=−5 are parallel or perpendicular. Explain.

Example 3 pg. 241 Parallel or Perpendicular lines Determine whether the graphs of 3𝑥+𝑦=12, 𝑦= 1 3 𝑥+2, and 2𝑥−6𝑦=−5 are parallel or perpendicular. Explain. 𝑦= 1 3 𝑥+2 is parallel to 2𝑥−6𝑦=−5, because they have equal slopes. 𝑥+3𝑦=12 is perpendicular to them both because the product of their slopes, 1 3 and −3 is −1.

Example 4 pg. 241 Perpendicular lines through a given point. Write an equation in slope-intercept form for the line that passes through 4, −1 and is perpendicular to the graph of 7𝑥−2𝑦=3

Example 4 pg. 241 Perpendicular lines through a given point. Write an equation in slope-intercept form for the line that passes through 4, −1 and is perpendicular to the graph of 7𝑥−2𝑦=3 𝑦=− 2 7 𝑥+ 1 7

4-4 pg. 243 11-37o, 39-41, 51-72(x3)

Scatter plots and lines of fit 4-5 Notes for Algebra 1 Scatter plots and lines of fit

Bivariate Data Data with two variables

Scatter Plot Used to show the relationship between a set of data with two variables, graphed as ordered pairs on a coordinate plane. They are used to investigate a relationship between two quantities.

Example 1 pg. 247 Evaluate a Correlation Computer Sharing in Maria’s School Students per computer 40 35 30 25 20 15 10 5 2000 2004 2008 2010 Year TECHNOLOGY the graph shows the average number of students per computer in Maria’s school. Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.

Example 1 pg. 247 Evaluate a Correlation Computer Sharing in Maria’s School Students per computer 40 35 30 25 20 15 10 5 2000 2004 2008 2010 Year TECHNOLOGY the graph shows the average number of students per computer in Maria’s school. Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. A negative correlation; each year, Maria’s school gets more computers, making the student-per-computer rate smaller

Line of Fit Trend line, a line that lies close to all points in a scatter plot.

Example 2 pg. 248 Write a line of Fit POPULATION The table shows the growth of the world population. Identify the independent and dependent variables. Make a scatter plot and determine what relationship, if any, exist in the data Year Population (millions) 1650 500 1850 1000 1930 2000 1975 4000 2004 6400

Example 2 pg. 248 Write a line of Fit POPULATION The table shows the growth of the world population. Identify the independent and dependent variables. Make a scatter plot and determine what relationship, if any, exist in the data Independent variable: year Dependent variable: pop. Positive Correlation 𝑦=35.1𝑥−63,870 Population (millions) World Population 7000 6000 5000 4000 3000 2000 1000 1500 1700 1900 2100 Year Year Population (millions) 1650 500 1850 1000 1930 2000 1975 4000 2004 6400

Linear extrapolation/interpolation Used to predict values outside/inside the range of data.

Example 3 pg. 249 Use Interpolation/Extrapolation Use the equation of the line of fit in Example 2 to predict the world’s population in 2015.

Example 3 pg. 249 Use Interpolation/Extrapolation Use the equation of the line of fit in Example 2 to predict the world’s population in 2015. 7,205.5 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

4-5 pg. 250 5, 7, 9, 18-42(x3)

Regression and Median-Fit Lines 4-6 Notes for Algebra 1 Regression and Median-Fit Lines

Best-fit-line Many Calculators use complex algorithms that find a more precise line of fit. Linear Regression—is one such algorithms Correlation coefficient—this number will tell you if the correlation is positive/negative and how closely the equation is modeling the data. (the closer the correlation coefficient is to 1 or −1 the more closely the equation models the data)

Example 1 pg. 255 Best fit line EARNINGS The table below shows Ariana’s hourly earnings for 2001- 2007. Use a graphing calculator to write an equation for the best-fit line for that data. Name the correlation coefficient. Let x be the number of years since 2000. Year Cost 2001 $10 2002 $10.50 2003 $11 2004 $13 2005 $15 2006 $15.75 2007 $16.50

Example 1 pg. 255 Best fit line EARNINGS The table below shows Ariana’s hourly earnings for 2001- 2007. Use a graphing calculator to write an equation for the best-fit line for that data. Name the correlation coefficient. Let x be the number of years since 2000. Year Cost 2001 $10 2002 $10.50 2003 $11 2004 $13 2005 $15 2006 $15.75 2007 $16.50

Example 1 pg. 255 Best fit line EARNINGS The table below shows Ariana’s hourly earnings for 2001- 2007. Use a graphing calculator to write an equation for the best-fit line for that data. Name the correlation coefficient. Let x be the number of years since 2000. 2nd → Catalog Alpha 𝒙 −𝟏 D Scroll down to DiagnosticOn Enter DONE STAT 1 Pulls up list under 𝑳 𝟏 1, 2, 3, 4, 5, 6, 7 𝑳 𝟐 10, 10.50, 11, 13, 15, 15.75, 16.50 Slide over to calculate 4 This will bring up the LinReg 𝒂𝒙+𝒃 Year Cost 2001 $10 2002 $10.50 2003 $11 2004 $13 2005 $15 2006 $15.75 2007 $16.50

Example 1 pg. 255 Best fit line EARNINGS The table below shows Ariana’s hourly earnings for 2001- 2007. Use a graphing calculator to write an equation for the best-fit line for that data. Name the correlation coefficient. Let x be the number of years since 2000. LinReg 𝑦=𝑎𝑥+𝑏 𝑎=1.214285714 𝑏=8.25 𝑟 2 = .9605317823 𝑟= .9800672336 𝑦=1.21𝑥+8.25; 𝑟=0.9801 2nd → Catalog 𝒙 −𝟏 D Scroll down to DiagnosticOn Enter DONE STAT 1 Pulls up list under 𝑳 𝟏 1, 2, 3, 4, 5, 6, 7 𝑳 𝟐 10, 10.50, 11, 13, 15, 15.75, 16.50 Slide over to calculate 4 LinReg 𝒂𝒙+𝒃 Answer Year Cost 2001 $10 2002 $10.50 2003 $11 2004 $13 2005 $15 2006 $15.75 2007 $16.50

Residual Measures how much the data deviates from the regression line. (The difference between the observed y-value and the predicted y- value <found on the best fit line>)

Example 2 pg. 256 Graph and Analyze a Residual Plot Graph and analyze the residual plot for the data comparing the years since the 2000 and the cost of repairs. Years Cost 2 1236 3 1560 4 1423 5 1740 6 2230

Example 2 pg. 256 Graph and Analyze a Residual Plot Graph and analyze the residual plot for the data comparing the years since the 2000 and the cost of repairs. Follow the steps for LinReg, then Years Cost 2 1236 3 1560 4 1423 5 1740 6 2230 2nd 𝒚= 2 → On Enter 𝒙−𝒍𝒊𝒔𝒕 𝑳 𝟏 𝒚−𝒍𝒊𝒔𝒕 STAT 7 RESID ZOOM 9

Example 2 pg. 256 Graph and Analyze a Residual Plot Graph and analyze the residual plot for the data comparing the years since the 2000 and the cost of repairs. Follow the steps for LinReg, then The plot appears to have a curved pattern, so the regression line may not fit the data well Years Cost 2 1236 3 1560 4 1423 5 1740 6 2230 2nd 𝒚= 2 → On Enter 𝒙−𝒍𝒊𝒔𝒕 𝑳 𝟏 𝒚−𝒍𝒊𝒔𝒕 STAT 7 RESID ZOOM 9 𝟎, 𝟕 𝐬𝐜𝐥:𝟏, by −𝟏𝟎𝟎𝟎, 𝟏𝟎𝟎𝟎 𝐬𝐜𝐥:𝟖𝟎𝟎

Example 3 pg. 257 Use Interpolation and Extrapolation BOWLING The table below shows the points earned by the top ten bowlers in a tournament. Estimate how many points the 15th-ranked bowler earned. Rank Score 1 210 2 197 3 164 4 158 5 151 6 147 7 144 8 142 9 134 10 132

Example 3 pg. 257 Use Interpolation and Extrapolation BOWLING The table below shows the points earned by the top ten bowlers in a tournament. Estimate how many points the 15th-ranked bowler earned. Follow the steps for example 1 Rank Score 1 210 2 197 3 164 4 158 5 151 6 147 7 144 8 142 9 134 10 132

Example 3 pg. 257 Use Interpolation and Extrapolation BOWLING The table below shows the points earned by the top ten bowlers in a tournament. Estimate how many points the 15th-ranked bowler earned. Follow the steps for example 1 𝑦=−7.87𝑥+201.2 Rank Score 1 210 2 197 3 164 4 158 5 151 6 147 7 144 8 142 9 134 10 132 Graph 𝒚= VARS 5 Statistics Slide EQ 1 2nd TRACE Enter 15

Example 3 pg. 257 Use Interpolation and Extrapolation BOWLING The table below shows the points earned by the top ten bowlers in a tournament. Estimate how many points the 15th-ranked bowler earned. Follow the steps for example 1 𝑦=−7.87𝑥+201.2 ≈83.1 Rank Score 1 210 2 197 3 164 4 158 5 151 6 147 7 144 8 142 9 134 10 132 Graph 𝒚= VARS 5 Statistics Slide EQ 1 2nd TRACE Enter 15

Median-Fit Line A line that is calculated using the medians of the coordinates of the data points

Example 4 pg. 258 Median-Fit line Find and graph the equation of a median-fit line for the data on the bowling tournament in Example 2. The predict the score of the 20th ranked bowler.

Example 4 pg. 258 Median-Fit line Find and graph the equation of a median-fit line for the data on the bowling tournament in Example 3. The predict the score of the 20th ranked bowler. Clear the 𝑦= list and graph the scatter plot. STAT Slide to CALC 3 ENTER

Example 4 pg. 258 Median-Fit line Find and graph the equation of a median-fit line for the data on the bowling tournament in Example 3. The predict the score of the 20th ranked bowler. Clear the 𝑦= list and graph the scatter plot. STAT Slide to CALC 3 ENTER 𝒚=−𝟗𝒙+𝟐𝟎𝟗.𝟓 Follow the steps for example 3, and enter 20

Example 4 pg. 258 Median-Fit line Find and graph the equation of a median-fit line for the data on the bowling tournament in Example 3. The predict the score of the 20th ranked bowler. Clear the 𝑦= list and graph the scatter plot. STAT Slide to CALC 3 ENTER 𝒚=−𝟗𝒙+𝟐𝟎𝟗.𝟓 Follow the steps for example 3, and enter 20 ≈𝟑𝟎

4-6 pg. 259 5, 7, 9, 21-42(x3)

Inverse Linear Functions 4-7 Notes for Algebra 1 Inverse Linear Functions

Inverse Relation A set of ordered pairs obtained by exchanging x-coordinates with the y- coordinates of each ordered pair in a relation.

Example 1 pg. 263 Inverse Relations Find the inverse of each relation. 1.) −3, 26 , 2, 11 , 6, −1 , −1, 20 2.) 𝒙 𝒚 −4 −3 −2 1 4.5 5 10.5

Example 1 pg. 263 Inverse Relations Find the inverse of each relation. 1.) −3, 26 , 2, 11 , 6, −1 , −1, 20 2.) 26, −3 , 11, 2 , −1, 6 , 20, −1 𝒙 𝒚 −4 −3 −2 1 4.5 5 10.5 𝒙 𝒚 −3 −4 −2 4.5 1 10.5 5

Example 2 pg. 264 Graph Inverse Relations Graph the inverse of each relation. 1.) 10 2.) 8 6 4 2 -8 -6 -4 -2 0000

Example 2 pg. 264 Graph Inverse Relations Graph the inverse of each relation. 1.) 10 2.) 8 6 4 2 -8 -6 -4 -2 0000

Inverse Function A linear relation that can be described as a function, has an inverse function.

Example 3 pg. 265 Find Inverse Linear Functions Find the inverse of each function. 1.) 𝑓 𝑥 =−3𝑥+27 2.) 𝑓 𝑥 = 5 4 𝑥−8

Example 3 pg. 265 Find Inverse Linear Functions Find the inverse of each function. 1.) 𝑓 𝑥 =−3𝑥+27 2.) 𝑓 𝑥 = 5 4 𝑥−8 𝑓 −1 𝑥 =− 1 3 𝑥+9 𝑓 −1 𝑥 = 4 5 𝑥+ 32 5

Example 4 pg. 266 Use an Inverse Function SALES Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings 𝑓 𝑥 for a month in which he compiled x dollars in total sales is 𝑓 𝑥 =2200+0.05𝑥. 1.) Find the inverse function. 2.) What do x and 𝑓 −1 𝑥 represent in the context of the inverse function? 3.) Find Carter’s total sales for last month if his earnings for that month were $3450.

Example 4 pg. 266 Use an Inverse Function SALES Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings 𝑓 𝑥 for a month in which he compiled x dollars in total sales is 𝑓 𝑥 =2200+0.05𝑥. 1.) Find the inverse function. 𝑓 −1 𝑥 =20𝑥−44,000 2.) What do x and 𝑓 −1 𝑥 represent in the context of the inverse function? x is Carter’s total earnings for the month, and 𝑓 −1 𝑥 is his total sales for the month 3.) Find Carter’s total sales for last month if his earnings for that month were $3450. $25,000

4-7 pg.267 9-35o, 45-60(x3)