Algebra 1 Notes Chapter 3

Graphing Linear Equations 3-1 Notes for Algebra 1 Graphing Linear Equations

Linear Equation An equation that forms a line when it graphed. Standard Form for a linear equation is 𝐴𝑥+𝐵𝑦=𝐶 C is called a constant (or a number) Ax and By are variable terms.

Example 1 pg. 155 Identify Linear Equations Determine whether each equation is a linear equation. Write the equation in standard form. 1.) 5𝑥+3=𝑥𝑦+2 2.) 3 4 𝑥=𝑦+8

Example 1 pg. 155 Identify Linear Equations Determine whether each equation is a linear equation. Write the equation in standard form. 1.) 5𝑥+3=𝑥𝑦+2 Not linear 2.) 3 4 𝑥=𝑦+8 Linear; 3𝑥−4𝑦=32

x-intercept The point where the line crosses the x-axis 𝑥, 0

y-intercept The point where the line crosses the y-axis. 0, 𝑦

Example 2 pg. 156 Find intercepts from a graph Find the x- and y-intercepts of the segment graphed below. 200 150 100 50 1 2 3 4

Example 2 pg. 156 Find intercepts from a graph Find the x- and y-intercepts of the segment graphed below. x-intercept is 4 y-intercept is 200 200 150 100 50 1 2 3 4

Example 3 pg. 157 Find intercepts from a table. ANALYZE TABLES A box of peanuts is poured into bags at a rate of 4 ounces per second. The table shows the function relating the weight of peanuts in the box and the time in seconds the peanuts have been pouring out of the box. 1.) Find the x- and y-intercepts of the graph of the function. 2.) Describe what the intercepts mean in terms of this situation. Pouring Peanuts Time(s) Weight (oz.) x y 2000 125 1500 250 1000 375 500

Example 3 pg. 157 Find intercepts from a table. ANALYZE TABLES A box of peanuts is poured into bags at a rate of 4 ounces per second. The table shows the function relating the weight of peanuts in the box and the time in seconds the peanuts have been pouring out of the box. 1.) Find the x- and y-intercepts of the graph of the function. x-int.=500, y-int.=2000 2.) Describe what the intercepts mean in terms of this situation. x-int.: 0 oz. after 500 s. y-int.: 2000 oz. before pouring began Pouring Peanuts Time(s) Weight (oz.) x y 2000 125 1500 250 1000 375 500

Example 4 pg. 157 Graph by using intercepts Graph 4𝑥−𝑦=4 by using the x- and y-intercepts.

Example 4 pg. 157 Graph by using intercepts Graph 4𝑥−𝑦=4 by using the x- and y-intercepts. 2 1 -1 3 4 5 -2 -3 -4

Example 5 pg. 158 Graph by making a table

Example 5 pg. 158 Graph by making a table 4 3 2 1 -3 -2 -1 -4 Graph 𝑦=2𝑥+2 x 𝟐𝒙+𝟐 y 𝒙,𝒚 -3 2 −3 +2 -4 −3,−4 2 0 +2 2 0, 2 3 2 3 +2 8 3, 8 6 2 6 +2 14 6, 14

3-1 pg. 159 13-49o, 42, 50, 51-55o, 66-75)(x3)

Solving Linear Equations by Graphing. 3-2 Notes for Algebra 1 Solving Linear Equations by Graphing.

Linear Function A function for which the graph is a line. The simplest linear function 𝑓 𝑥 =x is a called a parent function. A Family of Graphs is a group of graphs with one or more similar characteristics.

Solution/root Any value that makes an equation true. Zeros—values of x for which 𝑓 𝑥 =0 Zeros, Roots, solutions and x-intercepts are terms that represent the same thing.

Example 1 pg. 164 solve an Equation with one root. 1.) 0= 1 2 𝑥+3 2.) 2= 1 3 𝑥+3

Example 1 pg. 164 solve an Equation with one root. 1.) 0= 1 2 𝑥+3 -6 2.) 2= 1 3 𝑥+3 -3

Example 2 pg. 164 Solve an Equation with no solution 1.) 2𝑥+5=2𝑥+3 2.) 5𝑥−7=5𝑥+2

Example 2 pg. 164 Solve an Equation with no solution 1.) 2𝑥+5=2𝑥+3 2.) 5𝑥−7=5𝑥+2 − 2𝑥 −2𝑥 −5𝑥 −5𝑥 5≠3 No solution −7≠2 No solution

Example 3 pg. 165 Estimate by Graphing FUNDRAISING Kendra’s class is selling greeting cards to raise money for new soccer equipment. They paid $115 for the cards, and they are selling each card for $1.75. The function 𝑦=1.75𝑥−115 represents their profit y for selling x greeting cards. Find the zero of this function. Describe what this value means in this context.

Example 3 pg. 165 Estimate by Graphing FUNDRAISING Kendra’s class is selling greeting cards to raise money for new soccer equipment. They paid $115 for the cards, and they are selling each card for $1.75. The function 𝑦=1.75𝑥−115 represents their profit y for selling x greeting cards. Find the zero of this function. Describe what this value means in this context. ≈65.71; they must sell 66 cards to make a profit.

3-2 pg. 166 11-43o, 51-66(x3)

Rate of Change and Slope 3-3 Notes for Algebra 1 Rate of Change and Slope

Rate of Change A ratio that describes how much one quantity changes with respect to a change in another quantity. x—is the independent variable. y—is the dependent variable. 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 A positive rate of change indicates an increase over time. A negative rate of change indicates that a quantity is decreasing.

Example 1 pg. 172 Find the Rate of Change DRIVING TIME Use the table to find the rate of change. Then explain its meaning. Time Driving (h) Distance Traveled (mi) x Y 2 76 4 152 6 228

Example 1 pg. 172 Find the Rate of Change DRIVING TIME Use the table to find the rate of change. Then explain its meaning. 38 1 ; this means the car is traveling at a rate of 38 miles per hour. Time Driving (h) Distance Traveled (mi) x Y 2 76 4 152 6 228

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 1.) Find the rate of change for 2002-2004 9 and 2004-2006. 7 2.) Explain the meaning of the rate of change 5 in each case. 0 2002 2004 2006 3.) How are the different rates of change shown on the graph. 12.1 8.9 7.0

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 1.) Find the rate of change for 2002-2004 9 and 2004-2006. 7 5 0 2002 2004 2006 950,000/yr 1,600,000/yr 12.1 8.9 7.0

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 9 7 2.) Explain the meaning of the rate of change 5 in each case. 0 2002 2004 2006 For 2002-2004 there was an annual increase of 950,000 passports issued. Between 2004-2006, there was an average yearly increase of 1,600,000 passports issued. 12.1 8.9 7.0

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 9 7 5 0 2002 2004 2006 3.) How are the different rates of change shown on the graph. There is a greater vertical change for 2004-2006 than for 2002-2004. Therefore, the section of the graph for 2004-2006 is steeper. 12.1 8.9 7.0

Slope (m) The slope of a non-vertical line is the ratio of the change in the y- coordinates (rise) to the change in the x-coordinates (run) 𝑠𝑙𝑜𝑝𝑒= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 𝑥 1 , 𝑦 1 , 𝑥 2 , 𝑦 2 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦−𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥−𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1

Example 4 pg. 175 Positive, Negative and Zero slopes Find the slope of the line that passes through each pair of points. 1.) −3, 2 and 5, 5 2.) −3, −4 and −2, −8 3.) −3, 4 and 4, 4

Example 4 pg. 175 Positive, Negative and Zero slopes Find the slope of the line that passes through each pair of points. 1.) −3, 2 and 5, 5 𝑚= 3 8 2.) −3, −4 and −2, −8 𝑚=−4 3.) −3, 4 and 4, 4 𝑚=0

Example 5 pg. 176 Undefined Slope Find the slope of the line that passes through −2, −4 and −2, 3 .

Example 5 pg. 176 Undefined Slope Find the slope of the line that passes through −2, −4 and −2, 3 . Undefined

Slope Summary Positive Slope– goes uphill from left to right Negative Slope– goes downhill from left to right Slope of 0– horizontal line (y = #) Undefined Slope– vertical line (x = #)

Example 6 pg. 176 Find Coordinates given the slope. Find the value of r so that the line through 6, 3 and 𝑟, 2 has a slope of 1 2 .

Example 6 pg. 176 Find Coordinates given the slope. Find the value of r so that the line through 6, 3 and 𝑟, 2 has a slope of 1 2 . 𝑟=4

3-3 pg. 177 15-39o, 43-45, 54-66(x3)

3-4 Notes for Algebra 1 Direct Variation

Direct Variation Is described as an equation of the form 𝑦=𝑘𝑥, where 𝑘≠0. This is a constant rate of change. k is the constant of variation/proportionality

Example 1 pg. 182 Slope and Constant of Variation. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. 1.) 2.) y = -4x y = 2x 2 1 -2 -1 3 4 -3 -4 4 3 2 1 -2 -1 (0, 0) (1, 2) (0, 0) 0000 (1, -4)

Example 1 pg. 182 Slope and Constant of Variation. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. 1.) 2.) Constant of variation: 2; slope:2 Constant of Variation: -4; slope: -4 y = -4x y = 2x 2 1 -2 -1 3 4 -3 -4 4 3 2 1 -2 -1 (0, 0) (1, 2) (0, 0) 0000 (1, -4)

Example 2 pg. 183 Graph a Direct Variation

Example 2 pg. 183 Graph a Direct Variation 4 3 2 1 -3 -2 -1 Graph 𝑦=− 3 2 𝑥

Example 3 pg. 183 Write and solve a Direct Variation Equation Suppose y varies directly as x and 𝑦=9 when 𝑥=−3. 1.) Write a direct variation equation that relates x and y. 2.) Use the direct variation equation to find x when 𝑦=15.

Example 3 pg. 183 Write and solve a Direct Variation Equation Suppose y varies directly as x and 𝑦=9 when 𝑥=−3. 1.) Write a direct variation equation that relates x and y. 𝑦=−3𝑥 2.) Use the direct variation equation to find x when 𝑦=15. 𝑥=−5

Example 4 pg. 184 Estimate using Direct Variation TRAVEL the Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. 1.) Write a direct variation equation to find the distance d driven in time t. 2.) Graph the equation.

Example 4 pg. 184 Estimate using Direct Variation TRAVEL the Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. 1.) Write a direct variation equation to find the distance d driven in time t. 𝑑=60𝑡 2.) Graph the equation. 500 400 300 200 100 0 1 2 3 4 5 6

3-4 pg. 185 11-37o, 38-40, 42-44, 51-72(x3)

Arithmetic Sequences as Linear Functions 3-5 Notes for Algebra 1 Arithmetic Sequences as Linear Functions

Sequence A set of numbers (terms of the sequence) in a specific order.

Arithmetic sequence A sequence where the common difference between successive terms is a constant (called the common difference d)

Ellipsis (…) The 3 dots that are used with sequences to indicate there are more terms in the sequence.

Example 1 pg. 190 Identify Arithmetic Sequences Determine whether each sequence is an arithmetic sequence. Explain. 1.) −15, −13, −11, −9,… 2.) 7 8 , 5 8 , 1 8 , − 5 8 , …

Example 1 pg. 190 Identify Arithmetic Sequences Determine whether each sequence is an arithmetic sequence. Explain. 1.) −15, −13, −11, −9,… yes, there is a common difference of 2 2.) 7 8 , 5 8 , 1 8 , − 5 8 , … No, there is no common difference.

Example 2 pg. 190 Find the next term Find the next 3 terms of the arithmetic sequence −8, −11, −14, −17, …

Example 2 pg. 190 Find the next term Find the next 3 terms of the arithmetic sequence −8, −11, −14, −17, … −20, −23, −26

nth term of an Arithmetic Sequence Let 𝑎 1 be the 1st term in an arithmetic sequence, and d be the common difference. 2nd term-- 𝑎 2 =𝑎 1 +𝑑 3rd term-- 𝑎 3 =𝑎 1 +2𝑑 4th term-- 𝑎 4 =𝑎 1 +3𝑑 ⋮ ⋮ nth term-- 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑

Example 3 pg. 191 Find the nth term 1.) Write an equation for the nth term of the arithmetic sequence 1, 10, 19, 28, … 2.) Find the 12th term of the sequence. 3.) Graph the first 5 terms of the sequence 4.) Which term of the sequence is 172?

Example 3 pg. 191 Find the nth term 1.) Write an equation for the nth term of the arithmetic sequence 1, 10, 19, 28, … 𝑎 𝑛 =9𝑛−8 2.) Find the 12th term of the sequence. 100 3.) Graph the first 5 terms of the sequence 4.) Which term of the sequence is 172? 20th term 40 35 30 25 20 15 10 5 1 2 3 4 6

Example 4 pg. 192 Arithmetic Sequences as Functions NEWSPAPERS The Arithmetic sequence 12, 23, 34, 45, … represents the total number in ounces that a bag weighs after each additional newspaper is added. 1.) Write a function to represent this sequence. 2.) Graph the function and determine the domain.

Example 4 pg. 192 Arithmetic Sequences as Functions NEWSPAPERS The Arithmetic sequence 12, 23, 34, 45, … represents the total number in ounces that a bag weighs after each additional newspaper is added. 1.) Write a function to represent this sequence. 𝑎 𝑛 =11𝑛+1 2.) Graph the function and determine the domain. 𝐷=0, 1, 2, 3, … 50 45 40 35 30 25 20 15 10 5 -1 1 2 3 4 (4, 45) (3, 34) (2, 23) (1, 12)

3-5 pg. 193 9-23o, 24-30, 39-54(x3)

Proportional and Non-proportional Relationships 3-6 Notes for Algebra 1 Proportional and Non-proportional Relationships

Proportional Relationships If its equation is of the form 𝑦=𝑘𝑥 and 𝑘≠0. The graph will pass through 0, 0 .

Example 1 pg. 198 Proportional Relationships ENERGY The table shows the number of miles driven for each hour of driving. 1.) Graph the data. What can you deduce from the pattern about the relationship between the number of hours of driving h and the number of miles driven m. 2.) Write an equation to describe this relationship. 3.) Use this equation to predict the number of miles driven in 8 hours. Hours 1 2 3 4 Miles 50 100 150 200

Example 1 pg. 198 Proportional Relationships ENERGY The table shows the number of miles driven for each hour of driving. 1.) Graph the data. What can you deduce from the pattern about the relationship between the number of hours of driving h and the number of miles driven m. There is a linear relationship between the hours of driving and miles driven. Hours 1 2 3 4 Miles 50 100 150 200 250 200 150 100 50 1 2 3 4 5

Example 1 pg. 198 Proportional Relationships ENERGY The table shows the number of miles driven for each hour of driving. 2.) Write an equation to describe this relationship. 𝑚=50ℎ 3.) Use this equation to predict the number of miles driven in 8 hours. 400 miles Hours 1 2 3 4 Miles 50 100 150 200

Non-Proportional Relationships A linear function that does not pass through 0, 0

Example 2 pg. 199 Non-Proportional Relationships Write an equation in function notation for the graph. 16 14 12 10 8 6 4 2 1 3 5

Example 2 pg. 199 Non-Proportional Relationships Write an equation in function notation for the graph. 𝑓 𝑥 =3𝑥−2 16 14 12 10 8 6 4 2 1 3 5

3-6 pg. 200 5-13o, 21-33