The Coordinate Plane and Linear Equations Algebra 1

Transcription

1 Name: The Coordinate Plane and Linear Equations Algebra 1 Date: We use the Cartesian Coordinate plane to locate points in two-dimensional space. We can do this b measuring the directed distances the point lies awa from a horizontal and vertical ais. The first eercise and diagram will review some major features of the coordinate plane. Eercise #1: (a) On the diagram, the coordinates of point A are shown State the coordinates of points B, C, and D. B A ( 3, 5 ) (b) What are the coordinates of the origin, point O? II I (c) If a point is plotted in the first quadrant, I, then what is true about the signs of both its and coordinates? III C O IV D It is etremel important for ever Algebra student to be able to quickl and accuratel plot points when given in coordinate form. Practice this skill in Eercise #. Eercise #: Given the points A( 1, ), B( 3, 4 ), C (, 5 ), and D( 4, 6) the grid given below and state the quadrant that each point lies in., plot and label them on QUADRANTS A: B: C: D: Algebra 1, Unit # Linear Functions L1

2 Graphing a Linear Equation When we plot an equation, we are creating a picture, called a graph, of all coordinate pairs (, ) that are solutions to the equation. Graphs allow the Algebra student to quickl see man solutions to an equation. When we plot multiple points, sometimes these points will form a linear relationship; that is, when plotted, the solutions will form a line. The general form of a linear equation is shown below. GENERAL FORM OF A LINEAR EQUATION IN TWO VARIABLES A + B = C where A, B, and C are an real numbers such that A 0 and B 0. The following eercises will illustrate the important skill of plotting such linear equations. Eercise #3: Consider the linear relationship given b the equation = 4 3. (a) Create a table of values for this linear relationship. (b) Graph the linear equation on the aes below. 4 3 (, ) Keep in mind the big idea when graphing an equation, linear or otherwise: GRAPHING SOLUTIONS TO EQUATIONS If an ordered pair (, ) satisfies the equation (makes it a true statement), then it falls on the graph of that equation. Likewise, if a point (, ) falls on the graph of an equation, then it lies in the solution set of that equation. Thus a graph represents a picture of all solutions to an equation. Algebra 1, Unit # Linear Functions L1 The Arlington Algebra Project, LaGrangeville, NY, 1540

3 Eercise #4: Consider the linear equation + 4 = 10. (a) Solve this equation for so that it ma be entered into our graphing calculator. (b) Fill in the -chart below b creating a table on our calculator. 1 Y (, ) (c) Graph the linear equation on the aes below. The following sets of eercises will tr to reinforce the big idea of graphing b having ou work more with ordered pairs and linear equations. Eercise #5 Given the linear equation = 4 3 answer the following questions. (a) Does the point (, 5 ) satisf this equation? Justif. (b) Does the point (, 5 ) lie on the graph of this linear equation? Eplain. Eercise #6 Is the ordered pair ( 8,1) a solution to the equation 1 = 3? Justif. Algebra 1, Unit # Linear Functions L1 The Arlington Algebra Project, LaGrangeville, NY, 1540

4 Eercise #7 Given the equation satisfies the equation. 1 = 3, find the value of b given the fact that the point ( 0, b ) Eercise #8 Find the value of a such that the point (, 0) a lies on the graph of = + 8. Eercise #9 The graph of the equation 3 = 4 is shown at the right. Which of the following ordered pairs is not a solution to this equation? Eplain our choice. (1) (, 1) (3) ( 3, 4) () ( 0, ) (4) (, 5 ) Eplanation: Algebra 1, Unit # Linear Functions L1 The Arlington Algebra Project, LaGrangeville, NY, 1540

5 Name: The Coordinate Plane and Linear Equations Algebra 1 Homework Date: Skills 1. On the set of aes to the right, fill the quadrant numbers in the boes.. State which quadrant each of the following points lies. ( 3, ) ( 1, 3 ) ( 7, 5) ( 4, 6) O 3. State the coordinates of the origin, O. 4. Fill in the -chart below for the linear function = without the use of our calculator. Then, graph the equation on the aes to the right. - (, ) For each linear equation shown below, solve the equation for in terms of. (a) 3 + = 1 (b) 3 = (c) 4 + = 8 Algebra 1, Unit # Linear Functions L1

6 For problems 6 through 8, use our answers from problem 5 and our graphing calculator to create,-charts and plots of the linear equations = 1 1 Y (, ) 7. 3 = 1 Y (, ) = 8 1 Y (, ) Algebra 1, Unit # Linear Functions L1 The Arlington Algebra Project, LaGrangeville, NY, 1540

7 Reasoning 9. Two equations are shown below. One is linear and one is not. Which is the linear one and wh is it linear? 1 = + = Consider the linear equation 5 + = 3. (a) Does the point ( 1, 6) lie in the solution set of this equation? Justif. (b) Does the point ( 1, 6) lie on the graph of this linear equation? Eplain. (c) For what value of a will the point ( 1, a) fall on the graph of this linear equation? 11. The graph of a linear equation is shown on the aes below. Answer the following questions based on this graph. (a) For what value of b is the point ( 0, b ) a solution to this linear equation? Justif. (b) For what value of a will the point ( a, 3) be a solution to this linear equation? Justif. Algebra 1, Unit # Linear Functions L1 The Arlington Algebra Project, LaGrangeville, NY, 1540

8 Name: Slope and Parallel Lines Algebra 1 Date: The slope of a line measures the steepness of the line. We re familiar with the word slope as it relates to mountains. Skiers and snowboarders refer to hitting the slopes. Slope measures the ratio of the change in the -value of a line to a given change in its -value. DEFINITION OF SLOPE rise change in slope = m = = = run change in Slope is oftentimes smbolized using the variable m. Think of the slope of the line as the line s movement and this will help ou remember what it signifies. Eercise #1: For each of the following lines, state the slope, if it eists. (a) (b) rise m = run = rise m = run = (c) (d) rise m = run = rise m = run = The slope of a line is important because it tells us two things: (1) how steep the line is and () whether the line rises or falls as gets larger. Algebra 1, Unit # Linear Functions L

9 Eercise #: Below is a list of words. Fill in the blank of each statement below to make it true. Words ma be used more than once. undefined falls rises zero runs (a) When a line has a positive slope, it from left to right. (b) When a line has a negative slope, it from left to right. (c) When a line is horizontal, it onl from left to right and has slope of. (d) When a line is vertical, it onl and has an slope. It is important to be able to calculate the slope of a line if ou are given two points on that line. THE SLOPE FORMULA The slope of the line that passes through the points ( 1, 1) and (, ) 1 m = is given b 1 The subscripts (little numbers) just indicate that these are two different points. As seen in Eercises #1 and #, horizontal lines have slopes of zero and vertical lines have undefined slopes. Eercise #3: Find the slope of the line that passes through the points ( 8, 7 ) and ( 4, 5 ). Compute the slope using two different orders. What do ou notice? Order #1: ( 8, 7 ) and ( 4, 5 ) Order #: ( 4,5 ) and ( 8, 7 ) Eercise #4: The graphs of two lines are shown below. If etended these lines would not intersect at an points. (a) Calculate the slopes of both lines graphicall. a Line a Line b b (b) What are coplanar lines called that never intersect? Algebra 1, Unit # Linear Functions L

10 PARALLELISM AND SLOPE In a plane, two nonvertical lines are parallel if and onl if the have equal slopes. Of course, two vertical lines are parallel, but have noneistent slopes. Eercise #5: Is line AB parallel to line CD Justif. given A( 1,1 ), B ( 3, 7 ), C (, 0) and ( 5, 9) ( AB CD ) D. Eercise #6: Is AB CD given A(, 4 ), B( 6, 8 ), C ( 3, 5) and ( 9, 7) D. Justif. Eercise #7: Two linear functions are given below. = 1 and = (a) Enter these two equations into our graphing calculator and fill out the table shown at the right. (b) For both functions, b how much does each -value increase as the -variable increases b eactl one unit? Y Y (c) Eplain wh there is a constant difference of 4 units in the - values for the two functions. (d) Graph the two equations using the window shown at the right. How would ou characterize these two lines? min ma scl min ma scl = = 5 = 1 = = 10 = 1 Algebra 1, Unit # Linear Functions L

11 Name: Skills Slope and Parallel Lines Algebra 1 Homework Date: 1. Find the slope of the line that passes through each of the following sets of points. If the slope does not eist, so state. (a) ( 4, ) and ( 10, 8 ) (b) ( 0, 7 ) and ( 4, 7) (c) ( 4, 1 ) and (, 9) (c) (, 6 ) and (,13) (d) ( 1, 3 ) and ( 5, 3) (e) (, 8 ) and ( 6, 4). Find the slope of each of the following lines graphicall: (a) (b) (c) rise m = = run rise m = = run rise m = = run 3. The slope of line AB is. Line CD is parallel to line AB. Which of the following must be the slope of CD? (1) () (3) 1 (4) 1 passes through the points A( 4, ) and B ( 6, 4) 4. Line AB is the slope of CD? Justif.. Line AB is parallel to line CD. What Algebra 1, Unit # Linear Functions L

12 5. If PQ RS 1 and the slope of PQ = 4 algebraicall or numericall. 3 and the slope of RS is, then find the value of. Justif 8 Reasoning 6. Is line AB CD given the points A( 4, 3 ), B( 0, 5 ), C (,11 ), and D( 10, 7)? Justif. 7. Is line AB CD given the points A(, 4 ), B( 3, ), C ( 5, 6 ), and D( 7,10)? Justif The slope of AB is. AB value of. (The use of the grid is optional.) passes through the points A( 1, ) and B( 3, ). Determine the 1 9. The slope of a line is on the line? (The use of the grid is optional.) (1) (,1) (3) (, ) and contains the point ( 4, 3). Which of the following points also falls () ( 4, 1) (4) ( 1, 1) Algebra 1, Unit # Linear Functions L

13 Name: Slope as Rate of Change Algebra 1 Date: We know that slope measures the steepness of a line b comparing how much the -variable changes, its rise, to how much the -variable changes, its run, b using a ratio. We use ratios to compare the rates at which two quantities change. These comparisons are called rates of change. Eercise #1: Points A, B, and C are shown on the graph below. (a) Calculate the slope for each of the following line segments. Write our answer in reduced form. C AB BC AC B (b) Place point D at an point on line AC. Without calculation, what is the slope of AD? Eplain. A We see from this eercise that an two points along the line can be used to calculate the slope (rate of change) of the line. This is true for all sets of two variables that are linearl related (form the graph of a line). Eercise #: The following graphs shows Raquel s distance from home as she drives to college. (a) Determine the slope of this linear relationship using the slope formula and the two points that are shown Distance (miles) ( 6, 348 ) (b) What are the units of the slope? Hint Consider the units of the numerator and denominator. (c) What does this slope represent? (, 116) Time (hrs) SLOPE AS RATE OF CHANGE Algebra 1, Unit # Linear Functions L3 dependent slope = = rate of change independent

14 Eercise #3: Hachi was driving a car at a constant speed on the New York State Thruwa. He noticed that after driving for 1 hour he had 1 gallons of gas left. Then, after driving for a total of 3 hours, he had 8 gallons of gas left. Assume in this problem that there is a linear relationship between the amount of gasoline left and the time Hachi has been driving. (a) Epress the information given in this problem as two ordered pairs, where the independent variable is time and the dependent variable is the gallons of gasoline left. (b) Calculate the slope of the line connecting these two points. Include units in our answer. (c) How long, after he started driving, will Hachi run out of gasoline, assuming the rate ou calculated in part (b) does not change (is constant)? Eercise #4: Over the time interval given, calculate the rate of change for each variable. Include units in our answer: (a) At 3 weeks old, a corn plant is 4 inches tall and at 15 weeks it is 46 inches tall. (b) After driving for 4 hours, Tom is 8 miles from home and after driving for 7 hours, Tom is 44 miles from home. Unit Conversions One etremel important skill for both mathematics and science is the abilit to change from one set of units to another. All of these changes are done b multipling b the rate that one unit changes compared to another. Eercise #5: The space shuttle can fl at a speed of 5,000 feet in one second. How man times faster does the space shuttle fl compared to a car going 55 miles per hour? Use the fact that one mile is 580 feet long and round our final answer to the nearest integer. Algebra 1, Unit # Linear Functions L3

15 Name: Date: Slope as Rate of Change Algebra 1 Homework Applications 1. Over the time interval given, calculate the rate of change. Include appropriate units in each of our calculations. (a) A bo is 9 inches tall when he is 1 ear old and 66 inches tall when he is 15 ears old. (b) A car drives 80 miles in hours and 00 miles in 5 hours. (c) Takeru The Tsunami Kobashi eats 4 Nathan s hot dogs in minutes and 56 in 1 minutes. (d) Lebron James scored 6 points after 18 minutes and scored a total of 39 points b the end of the game (48 minutes).. Baden loans $100 to Kaile who pas it back at a rate of $10 per week. The amount of mone that is owed to Baden b Kaile is a linear function of time. What is the slope? Include units. 3. Admiral Al needs a ramp to perform a stunt at the Arlington High School pep rall. For the first 0 feet along the ground, the ramp must rise off of the ground at a constant rate. After the construction crew has finished 5 feet of the horizontal distance, the track is 4 feet off the ground. After 0 feet of horizontal distance is constructed, Admiral Al needs to clear a 1 foot high barrier. Will he be able to do it? Use of the grid paper below is optional. Algebra 1, Unit # Linear Functions L3

16 4. Elana is scaling up a 500-foot cliff. She starts at the bottom at t = 0 hours and carefull climbs at a constant rate of 10 feet per hour. (a) After 3 hours, has Elana reached the top of the cliff? Justif. (b) After how man minutes will Elana reach the top of the cliff? Show how ou arrived at our answer. (c) Epress our answer to part (b) in terms of hours and minutes. 5. For homecoming, the student government bought balloons. Feli accidentl lets one of the balloons go that is attached to a 400 foot string. The balloon rises at a rate of 7 feet per second. (a) Feli doesn t notice that he s let go of the balloon until 35 seconds have passed since it started to float upwards. Can he still catch the balloon? Justif. (b) Determine the time, to the nearest tenth of a second, when the string will run out. 6. Gabe takes a bike ride starting from home. He travels 4 miles in 6 minutes. He then gets stopped in traffic for 3 minutes. Frustrated, Gabe decides to bike home, which takes 8 more minutes. (a) Draw a graph of Gabe s distance from home as a function of the time since he left. (b) Find the slope of each portion of this graph. Include units Distance from Home (miles) Time Since Leaving (minutes) Algebra 1, Unit # Linear Functions L3

17 Name: Writing Equations of Lines Algebra 1 Date: It is important to be able to move between the graphical and algebraic forms of a line. The following eercise begins this process. Eercise #1: Consider the linear function given b the equation 3 = 3. (a) Using our calculator to generate an -chart, plot the function on the grid at the right. (b) Determine the slope of this line graphicall. (c) Determine the -intercept of this line graphicall. THE SLOPE-INTERCEPT FORM OF A LINE = m + b m = slope and b = the -intercept A good wa to think about the -intercept, b, is that it is where a line begins on the -ais. A good wa to think about the slope, m, is it gives the movement of the line. Eercise #: Write the equation, in = m + b form, for each line shown below. (a) (b) Equation: Equation: Algebra 1, Unit # Linear Functions L4

18 Writing Equations of Lines We want to develop the skill of writing equations of lines, in = m + b form, using a variet of information. In each of these problems, though, the end-goal is the same, to determine the value for the slope and the value of the -intercept. Eercise #3: Write the equation of a line in = m + b form that is parallel to the line = + 4 and has a -intercept of 8. Eercise #4: Write the equation of the line passing through the point (1, 8) with a slope of 3. Eercise #5: Consider a line that passes through the points (4, 10) and (6, 11). (a) Find the line s slope. (b) Find the line s -intercept algebraicall. (c) Write the equation for the line in = m + b form. (d) Using a table on our graphing calculator, verif that the two points given fall on this line. (e) Using our calculator table, determine the -coordinate of this line when = 4. (f) Using our calculator table, determine the -coordinate on the line when = 30. Eercise #6: Line a is shown on the grid below. (a) On the same set of aes, sketch the line that passes through point A(4, 3) and is parallel to line a. (b) Write the equation of the line that ou just drew. a A Algebra 1, Unit # Linear Functions L4

19 Name: Writing Equations of Lines Algebra 1 Homework Date: Skills 1. Write the equation of the line, in = m + b form, for each set of information given. (a) a slope of 3 and a -intercept of 9 (b) a slope of 1 and a -intercept of 0 (c) a slope of 6 and a -intercept of 3 (d) a slope of 0 and a -intercept of. Which of the following lines is parallel to a line whose equation is = ? Hint Arrange this line in its = m + b form first. (1) = + 5 (3) () = 3 + (4) 3 = 7 = Which of the following lines has a slope of 5 and a -intercept of 3? (1) = 5 3 (3) = () = 3 (4) = Which of the following is the equation for the graph shown at the right? (1) = + (3) 3 3 = () = (4) = 3 5. Which of the following is the -intercept of the line whose slope is 4 and which passes through the point ( 8,15 )? (1) 15 (3) 8 () 17 (4) 3 Algebra 1, Unit # Linear Functions L4

20 6. Which of the following lines is parallel to the line = and has a -intercept of 10? (1) = (3) = () = 3 10 (4) = Write the equation of the line, in = m + b form, that passes through the points ( 4, 3 ) and (, 6). The use of the grid is optional. 8. Write the equation of the line, in m b. = + form, that passes through the points ( 1, 4 ) and ( 3, ) The use of the grid is optional. Beware, the -intercept of this line is not an integer. 9. Write the equations of the lines that are parallel to the line = and pass through the following points. (a) ( 0, 7 ) (b) ( 5, 6 ) Algebra 1, Unit # Linear Functions L4

21 Name: Graphing Lines Algebra 1 Date: We now know the importance of the slope of a line. It gives us both how steep the line is and whether the line increases or decreases. Given the slope and a single point on a line, we can find all other points that lie on it as well. The net eercise will illustrate this concept. Eercise #1: For each part of this problem, the slope of a line is given along with a point through which the line passes. Graph the line and write its equation in = m + b form. 1 and 4, 3 and,1 (a) m = ( ) (b) m = ( ) EQUATION: EQUATION: We see from this eample that if we know a single point on a line and the line s slope, we can quickl graph the line. One point that is easil determined from its equation comes from the -intercept. Eercise #: Consider the linear equation 1 = 1. 3 (a) State the slope and the -intercept of this line. (b) Use the slope and the -intercept to graph the line to the right. Algebra 1, Unit # Linear Functions L5

22 Keep in mind that, in general, we can alwas create a table of values, either b hand or in our calculator, to graph an line. Using the slope and the -intercept is a quick wa to graph man lines. Of course, the ke in using these is to have the equation of the line in its = m + b form. Eercise #3: Consider the linear function 3 = 6. (a) Place this equation in = m + b form. (b) Graph the equation on the grid to the right. (c) Does the point ( 36, ) lie on this line? Justif. Eercise #4: For each of the following, rearrange into = m + b form and then graph. (a) = 0 (b) + = 4 (c) 3 4 = 4 (d) + = 8 Algebra 1, Unit # Linear Functions L5

23 Name: Graphing Lines Algebra 1 Homework Date: Skills 1. Which of the following is true of the line whose equation is + 3 = 8? (1) It has a slope of 3 and a -intercept of 8. () It has a slope of 3 and a -intercept of 8. (3) It has a slope of (4) It has a slope of 3 3 and a -intercept of 4. and a -intercept of 4.. Which of the following equations represents the graph shown? (1) 3 = 3 (3) = 3 3 () 3 = (4) = 3 3. Consider the linear function 1 = 1. (a) Graph this line on the grid using its slope and -intercept. (b) None of the following points, when plotted, would fit on this particular grid, but that doesn t mean the don t lie on the line (it etends forever in both directions). Determine which points lie on the line and which do not. ( 30,14 ) ( 70, 8 ) ( 30, 15) ( 50, 6) Algebra 1, Unit # Linear Functions L5

24 4. For each of the following linear equations, rearrange into = m + b form and then graph on the grid. (a) + = 4 (b) = 4 (c) = (d) = Reasoning 5. Consider the three linear graphs shown below as ou answer the following: (a) State the coordinates of the point for each graph s intersection with the -ais. (b) State the -intercept for each graph. (c) To determine the -intercept algebraicall, what number must be substituted for? Algebra 1, Unit # Linear Functions L5

25 Name: Horizontal and Vertical Lines Algebra 1 Date: There are two tpes of lines that are slightl different from the tpical slant line. These lines are horizontal, parallel to the -ais, and vertical, parallel to the -ais. Horizontal Line Vertical Line Eercise #1: The line = 3 is graphed on the grid at the right. (a) Graph and label the lines = 5 and = on the same grid. (b) State the coordinates of point A and B from the line = 3. (c) If it eists, find the slope of the line connecting points A and B from above. A B = 3 (d) If it eits, find the -intercept of the line connecting points A and B. (d) Write the equation of the line connecting A and B in = m + b form. EQUATIONS OF HORIZONTAL LINES = m + b m = = where 0 (or simpl b) Eercise #: Which of the following represents the equation of the graph shown at the right? (1) = 4 (3) = 4 () = 4 (4) = 4 Algebra 1, Unit # Linear Functions L6

26 Eercise #3: The line = is graphed on the grid at the right. = (a) Graph and label the lines = 4 and = 3 on the same grid. (b) State the coordinates of point A and B from the line =. B (c) If it eists, find the slope of the line connecting points A and B from above. (d) If it eists, find the -intercept of the line connecting points A and B. A (e) Wh is it not possible to write the equation of a vertical line in = m + b form? Eercise #4 Graph the following three lines and find the area of the triangle enclosed b them. = 5 = 3 = 1 EQUATIONS OF VERTICAL LINES = a a intercept where is the of the line Eercise #5 Create a rough sketch and then write the equation of the line that fits each description: (a) parallel to the -ais passing through ( 3, ) (b) parallel to the -ais passing through ( 4, 3) Algebra 1, Unit # Linear Functions L6

27 Name: Skills Horizontal and Vertical Lines Algebra 1 Homework 1. Which of the following equations represents the line shown in the graph to the right? Date: (1) = 3 () = 3 (3) = 3 (4) = 3. Which of the following equations represents the line shown in the graph to the right? (1) = () = (3) = (4) = 3. Graph and label the following two lines. Write the coordinates of their intersection point. = 5 = 4 Algebra 1, Unit # Linear Functions L6

28 4. Graph and label the following vertical and horizontal lines. Then, determine the area of the rectangle enclosed b the lines. = = 4 = 3 = 5 5. Graph and label the following lines. Then, determine the area of the triangle enclosed b the lines. Remember to solve for in the equation of the slant line to use our graphing calculator to set up a table. = 3 = 3 = 6 6. Which of the following represents the equation of the -ais? (1) = 1 (3) = 0 () = 1 (4) = 0 7. Which of the following represents the equation of a line that is parallel to the -ais and passes through the point (1, 4)? (1) = 1 (3) = 4 () = 4 (4) = 1 8. Which of the following equations represents a line that is parallel to the -ais and passes through 3, 5? the point ( ) (1) = 3 (3) = 3 () = 5 (4) = 5 Algebra 1, Unit # Linear Functions L6

29 Name: Modeling with Linear Functions Algebra 1 Date: We have learned that slope is used to describe real life rates of change. We also know that the - intercept is where the line begins on the -ais. The -intercept alwas occurs where the independent variable has a value of zero. Using these two quantities, slope and -intercept, we can model and solve man real life problems. Eercise #1: The Arlington Freshmen class wants to have a fundraiser. The class wants to bu a number of $4 flip-flops and $5 bracelets. The class has a total of $100 to spend. (a) If represents the number of flip-flops and represents the number of bracelets, complete the table below. # of flip-flops, 0 # of bracelets, 0 (b) Using the two points from part (a), write a linear equation in = m + b form that gives the number of bracelets that can be bought as a function of the number of flip-flops bought. (c) Using our equation from (b), determine the number of bracelets that can be bought if 10 flip-flops were purchased. Eercise #: From 000 to 007 the number of coffee shops in a certain countr increased b 100 shops per ear. In 00, there were 1100 coffee shops. (a) Write a linear equation for the number of coffee shops,, as a function of time, t, where t = 0 represents the ear 000. (b) Based on our linear model from part (a), predict the number of coffee shops that will be in that countr in 05. Algebra 1, Unit # Linear Functions L7

30 Eercise #3: The cost to subscribe to an online internet service consists of a $15 per month flat-fee and a $4.00 per hour additional charge. Cost of Subscription, C (a) Create a linear model to represent the total 100 cost per month, C, as a function of the number of hours, h, that are used. 75 (b) Using our calculator to generate a table of values, graph the model ou formed in part (a) on the grid provided. (c) Luc was charged $75 after signing up and using the service for one month. How man hours did she use? Justif our answer both algebraicall and graphicall Number of Hours Used, h Eercise #4: Shirle s Workout Shack charges $6 to sign up and $3 each time a person works out. (a) Write an equation representing the cost, C 1, to workout at Shirle s as a function of the number of workouts a person has worked out, w Total Cost, C (b) Tomm s Pump Up Center charges $14 to sign up and $ for each workout. Create another linear function, as in part (a), for the cost, C, of attending Tomm s Center. 0 (c) Graph both equations on the grid to the right. What number of workouts will result in the same cost for both gms? 10 Algebra 1, Unit # Linear Functions L Number of Workouts, w

31 Name: Modeling with Linear Functions Algebra 1 Homework Date: Applications 1. Hamal Rental Cars charges a flat-fee of $5 per da to rent a new Chev Impala, plus a mileage charge of $0.5 per mile. Cost of Renting, C (a) Write a linear equation to represent the cost, C 1, of renting an Impala as a function of the number of 35 miles driven, m. 30 (b) On the grid to the right, graph and label the linear function our created in part (a). (c) Ike s Rentals charges a flat fee of $0 per da to rent an Impala plus a mileage charge of $0.50 per mile. As in (a), write a linear equation to represent the cost, C, of renting an Impala from Ike s and graph this function on the grid at the right Number of Miles Driven, m (d) For what number of miles, m, will the rental costs be equal for the two places?. Kael wants to install a new toilet. Luigi the plumber charges $100 for the cost of the toilet plus an additional $75 per hour. (a) Write an linear equation that gives the cost, C 1, as a function of the hours, h, that Luigi works. (b) Being ver eact with his hours, Luigi charges Kael $750. Determine, to the nearest tenth of an hour, how long Luigi worked on this job. Justif our answer using algebra or tables in our calculator. If ou justif using a table, write at least three rows from our table. Algebra 1, Unit # Linear Functions L7

32 3. Javier is tring to find a linear equation for the cost of his cell-phone plan. The first month he talks for onl 3 minutes and is charged $ The second month he talks for 40 minutes and is charged $ (a) Write two ordered pairs, where the minutes are the independent variable and the charge is the dependent variable, that model the information given in the problem. (b) Using these two points, write a linear equation that gives Javier s charge, C, as a function of the number of minutes, m, that he talks. (c) What does the slope of this linear function represent? 4. Miguel is driving towards New York Cit at a constant rate of speed. After hours he notices that he is 17 miles awa and after 3 hours he notices that he is 69 miles awa. (a) Write the information above as two ordered pairs, with time being the independent variable and the distance from New York Cit being the dependent variable. (b) Using our ordered pairs from part (a), write a linear equation in which the distance Miguel is awa, D, as a function of the time he has been driving, t. (c) Wh is the slope of our linear equation from part (b) negative? Eplain in terms of the real-life scenario that the linear equation is modeling. (d) How far from NYC was Miguel when he started his trip at t = 0 hours? Justif. Algebra 1, Unit # Linear Functions L7

33 Name: Date: Solving a Sstem of Linear Equations Graphicall Algebra 1 If two or more equation are given, we call this a sstem of equations. The solution to a sstem of,, that satisf (make true) all of the equations in equations consists of the set of all ordered pairs, ( ) the sstem. In toda s lesson, we will investigate was of finding this solution set for two linear equations. Eercise #1: Consider the sstem of linear equations shown below. The graphs of both linear functions are shown on the grid at the right. = + = 4 (a) What point lies on both lines? = + = 4 (b) Algebraicall justif that the point from part (a) is the solution to this sstem of equations b checking to see if the point satisfies both equations. Eercise #: Solve the following sstem of linear equations b graphing each line using the slope and -intercept method. Then, check our solution. + 5 = 0 = Algebra 1, Unit # Linear Functions L8

34 Eercise #3: Which of the following is a solution to the sstem of equations consisting of = and = + 1? (1) ( 0,11 ) (3) (, 3) () ( 3, ) (4) (, 5 ) Eercise #4: Use our graphing calculator to set up an -chart for the following two linear functions. Then, solve the sstem of linear equations. = = 0 Eercise #5: Alice s Athletic Arena requires members to pa $0 to join and members must pa $1.50 for each time the come to work out. Ro s Romper Room requires members to pa $5 to join and members must pa $4 for each time the come to work out. Cost, C (a) Set up two linear functions for the cost, C, of working out at each gm as a function of the number of times, n, that a person works out. 40 CA = C = R 30 (b) Use our graphing calculator to help ou graph and label both functions on the grid to the right. Show our table below (c) For how man visits, n, will the cost at both gms be the same? Algebra 1, Unit # Linear Functions L Number of Visits, n

35 Name: Skills Solving a Sstem of Linear Equations Graphicall Algebra 1 Homework Date: For problems 1 through 3, solve each sstem graphicall b using the slope and the -intercept method to quickl graph our lines. Then, check our solutions with the aid of our calculator. 1. = 4 = + 5. = = = = + 6 Algebra 1, Unit # Linear Functions L8

36 4. If drawn on the same graph, the lines with equations = 3 and = + 18 would intersect at which of the following points? (1) ( 1,1 ) (3) ( 0,18 ) () ( 4,10 ) (4) ( 3,1 ) 5. Use our graphing calculator to set up an -chart for the following two linear functions. Then, solve the sstem of linear equations. + 3 = 7 3 = 10 Reasoning 6. Consider the linear sstem of equations shown to the right. (a) Sketch a graph of this sstem on the aes to the right. 3 = + 3 = 1 (b) Will these lines ever intersect? Justif. (c) Considering our answer from part (b), eplain wh the solution set of this sstem is empt (contains no ordered pairs). A sstem such as this one that has no solution is called an inconsistent sstem. Algebra 1, Unit # Linear Functions L8

37 Name: Lines of Best Fit Linear Regression Algebra 1 Date: When mathematical models are used in the real world, we often don t have data that fall on perfect lines. Most of the time we want to find the line of best fit (or the line that best fits the data). Eercise #1: A biologist is studing the relationship between a tree s diameter and its height. She records the following data for 7 different trees. Diameter (inches) Height (feet) (a) On the grid provided, create a scatterplot of the data. Use the diameter as the independent variable and the height as the dependent variable. (b) Draw a line of best fit through the data. As a guide, tr to have as man points of data fall above the line as below the line. (c) Write two ordered pairs that lie on our line. (d) Determine the equation of our linear function using the two ordered pairs from part (c). (e) Using our linear function from part (d), estimate, to the nearest foot, the height of a tree given that its diameter is 14 inches. (This tpe of calculation is called etrapolating; we are using a model to predict outside of our data set.) Algebra 1, Unit # Linear Functions L9

38 Entering Data in Your Calculator We will enter data frequentl in the calculator. Toda we will enter it in order to produce the equation of the line of best fit, which will certainl fit the data better than an rough estimate that we create b hand. Eercise #: Consider the same data that ou had before: Diameter (inches) Height (feet) (a) Enter the data into our calculator as follows. Step 1 Hit the STAT button and go to the Edit submenu. Step Enter the Diameters under L1 and the Heights under L. When working with a data set, alwas place the independent variable in L1 and the dependent variable in L. (b) Find the equation for the line of best fit. Round our coefficients to the nearest tenth. Also, define what each variable, and, represent. Step 1 Hit the STAT button and go to the CALC submenu. Step Go the choice 4 LinReg(a+b). Hit ENTER twice. (c) To the nearest foot, use our equation to find the height of a tree whose diameter is 14 inches. (d) Construct a table on our calculator to find the diameter of a tree, to the nearest tenth of an inch, for which the tree would be 45 feet tall. Provide numerical evidence to support our answer. Algebra 1, Unit # Linear Functions L9

39 Name: Lines of Best Fit Linear Regression Algebra 1 Homework Date: 1. The table below shows the number of students in Arlington High School as a function of the number of ears since 000. Number of Years Since 000, Number of Students at Arlington High School, (a) On the grid below, draw a scatterplot of the data. Population of Arlington High School 3400 (b) Determine an equation, using our calculator, for the line of best fit. Do not round our coefficients (c) Define below what each variable represents: represents: represents: Number of Years Since 000 (d) Use our model to predict the population of Arlington High School in the ear 01. Round our answer to the nearest whole number. (e) Use our model and a table on our graphing calculator to determine between which two consecutive whole number ears the population reaches Provide numerical evidence to support our answer. Algebra 1, Unit # Linear Functions L9

40 . A real-estate agent is tring to determine the relationship between the distance a 3-bedroom home is from New York Cit and its average selling price. He records data for 6 homes shown below. Miles from New York Cit, Price of 3 Bedroom Home, , , , , ,000 85,000 (a) Using our calculator, write a linear regression equation that relates the distance from New York Cit,, to the price of the 3-bedroom home,. Round our coefficients to the nearest hundred. (b) Woodstock, New York, is located 95 miles from New York Cit. Using our linear model from part (a), determine the price of a 3-bedroom home in Woodstock. (c) Using our model, determine the price of a 3-bedroom home in New York Cit. (Hint: think about the value of when ou are in New York Cit.) (d) Using tables, determine, to the nearest mile, the distance from New York Cit a home would be if its selling price were eactl $500,000. Provide numerical evidence to support our answer. (e) Using tables, determine, to the nearest mile, the distance from New York Cit a home would be if its selling price were $0. Provide numerical evidence to support our answer. (f) Wh is our answer to part (e) unreasonable? Eplain. When we use etrapolation with linear models we can sometimes get unreasonable answers. This is because we are using the model with independent variable values for which the model does not appl. Algebra 1, Unit # Linear Functions L9

41 Name: Correlation Coefficient Algebra 1 Date: In the last lesson we saw that for ever data set relating two variables a line of best fit can be found. How well this line fits the data given, though, is questionable. To measure this fit, we use what is known as the correlation coefficient, commonl given the letter r. Eercise #1: Turn our correlation coefficient on b doing the following steps. Step 1 Go to CATALOG and select DiagnosticsOn. Step Hit ENTER twice Eercise #: Consider the data set given to the right. (a) Graph the ordered pairs on the given grid. {( 4, ), (, 0 ),( 1, 3 ), ( 3,5) } (b) Determine the equation for the line of best fit. (c) State the value of the correlation coefficient. Eercise #3: Consider the data set given to the right. (a) Graph the ordered pairs on the given grid. (b) Determine the equation for the line of best fit. {(, 6 ), ( 0, 3 ),(, 0 ), ( 4, 3) } (c) State the value of the correlation coefficient. Algebra 1, Unit # Linear Functions L10 The Arlington Algebra Project, LaGrangeville, NY, 1540

42 Eercise #4: Consider the data set given to the right. (a) Graph the ordered pairs on the given grid. (b) Determine the equation for the line of best fit. Round coefficients to the nearest hundredth. Draw a rough sketch of the line of best fit on the aes. {(1,3),(3, 4),(5,6),(6,5) } (c) State the value of the correlation coefficient. answer to the nearest hundredth. Round our (d) Wh is the correlation coefficient not equal to one? Eercise #5: Consider the data set given to the right. {( 4, 5 ), ( 0, 4 ),(, ), ( 5, 4) } (a) Graph the ordered pairs on the given grid. (b) Determine the equation for the line of best fit. Round coefficients to the nearest tenth. Draw a rough sketch of the line of best fit on the aes. (c) State the value of the correlation coefficient. answer to the nearest hundredth. Round our (d) Wh is the correlation coefficient negative? CORRELATION COEFFICIENT SUMMARY, r-value (1) r has a value on the interval 1 r 1. () r = ± 1 indicates a perfect linear fit (3) r = 0 indicates no linear correlation (4) the sign of r indicates a positive or negative slope Algebra 1, Unit # Linear Functions L10

43 Name: Skills Correlation Coefficient Algebra 1 Homework Date: 1. For each of the si scatterplots drawn below, sketch the line of best fit. Then, for each choose the most appropriate r-value given from the list below all of the figures. (a) (b) (c) r = r = r = (d) (e) (f) r = r = r = Possible r-values: -1.00, -0.65, 0.00, 0.75, 0.9, Which of the following is closest to the correlation coefficient of the data set given below? (1) 0.7 (3) 0.95 () 0.89 (4) 0.5 Algebra 1, Unit # Linear Functions L10 The Arlington Algebra Project, LaGrangeville, NY, 1540

44 3. Find the correlation coefficient, r, to the nearest hundredths place, for each list of ordered pairs. (a) (-, 1), (0, 3), (, 5), (4, 7) r = (b) (-7, 10), (-5, 6), (-3, 8), (-1, ) r = (c) (-8, 8), (, -1), (-6, 8), (-4, 0), (3, -10), (1, 1) r = Applications 4. The low and high temperatures for one week in Poughkeepsie, New York, are given below. Low Temp, High Temp, (a) Write the best linear model, in = a + b form, that gives the high temperature as a function of the low temperature. Round all coefficients to the nearest tenth. (b) State the correlation coefficient of this data set to the nearest hundredth. (c) If this model held for the following week, what was the high on a da when the low temperature was 44? Round our answer to the nearest whole degree. (d) Another linear model predicted the high temperature based on the highest humidit of the da. For this linear model, the correlation coefficient was Which model does a better job in predicting the high temperature? Justif. Reasoning 5. The linear correlation coefficient provides two pieces of information about a data set and its line of best fit. What are the? Eplain. Algebra 1, Unit # Linear Functions L10

45 Name: Absolute Value Algebra 1 Date: In Unit #1, we talked about absolute value in terms of the distance a number is awa from zero on a number line. We will now investigate the graph of the absolute value function. Eercise #1: Consider the function =. (a) Without the use of our calculator, fill in the table below for this function. = (b) Graph the function on the grid below (c) Find the slope of the graph for each of the intervals stated below: < 0 > 0 Eercise #: Fill in the two blanks in the definition bo below. ALGEBRAIC DEFINITION OF ABSOLUTE VALUE 0 = = < 0 Eercise #3: Use the definition above to find the absolute value in each case below. (a) 5 = (b) 1 = (c) 0 = Algebra 1, Unit # Linear Functions L11

46 Eercise #4: Consider the function = + 3. (a) Using our graphing calculator to generate an -table, graph this function on the given grid. (b) State the coordinates of the turning point of this absolute value function. (c) How was the graph of this function? = shifted to produce the graph of Eercise #5: Consider the function = (a) Using our graphing calculator to generate an -table, graph this function on the given grid. (b) State the coordinates of the turning point of this absolute value function. (c) How was the graph of this function? = shifted to produce the graph of Eercise #6: Consider the function =. (a) Using a calculator to generate an -table, graph the function on the grid to the right. (b) How has the graph of graph of =? = been transformed to form the Algebra 1, Unit # Linear Functions L11

47 Name: Skills Absolute Value Algebra 1 Homework Date: 1. Which equation describes the graph shown below? (1) = + 5 (3) = 5 () = (4) = 5. Which equation describes the graph shown below? (1) = + 3 (3) = 3 () = 3 (4) = Lorraine entered an absolute value function in her graphing calculator and produced the table shown below. What are the coordinates of the turning point of this absolute value function? (1) ( 1,1 ) (3) ( 3, 5) () ( 7, 1) (4) ( 5, 3) 4. Consider the function = 4 3. (a) What are the coordinates of the turning point of this function? (b) Graph the function on the grid provided. Place our aes on the grid in such a wa that ou graph at least 5 integer - values greater than and less than the turning point s - coordinate. Algebra 1, Unit # Linear Functions L11

48 Applications 5. The roof of a tent above flat ground can be modeled b the equation = + 4, where represents the horizontal position from the center of the tent and represents the tent s height. Both variables have units of feet. (a) Graph this equation on the grid to the right. Onl graph that portion that represents the tent s height at ground level or above. (b) What is the maimum height of the roof above the ground? Include units. (c) What is the width of the tent? Include units. Reasoning 6. Consider the absolute value functions = and = (a) Enter these functions as Y1 and Y, respectivel, in our calculator. (b) Graph these functions b using the STANDARD ZOOM WINDOW. (c) The graph of onl one of these two functions is shown in this viewing window. Write the equation of the one that is drawn. (d) Eplain wh the graph of the other absolute value function does not appear in this viewing window. Algebra 1, Unit # Linear Functions L11