UNIT 9 Analtic Geometr An air traffi c controller uses algebra and geometr to help airplanes get from one point to another. 00 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
A pilot uses numbers to locate the airport she is fling to. An air traffic controller uses numbers on a radar screen to locate each airplane approaching the airport. Without a sstem of locating points, airplanes would have a hard time getting anwhere safel. Big Idea Analtic geometr combines the tools of algebra with geometr to solve man realworld problems. Unit Topics Points on the Plane Two-Variable Equations Linear Equations and Intercepts Slope Applications: Linear Graphs Relations and Functions Sstems of Linear Equations ANALYTIC GEOMETRY 0 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Points on the Plane You can use a single number to describe the location of a point on a line, but on a coordinate plane it takes two numbers to describe the location of an point. Identifing Aes and Quadrants A coordinate plane is formed b two perpendicular number lines called aes. The -ais is a horizontal line. The -ais is a vertical line. The aes intersect at the point at which the both have coordinate zero. This point is called the origin. The aes separate the plane into four quadrants. On the aes, positive goes right and positive goes up. Negative goes left and negative goes down. -ais Quadrant II the origin Quadrant III Quadrant I -ais O Quadrant IV TIP The quadrants are numbered counterclockwise. To identif Quadrant I, think of where the numeral is on a clock. Eample For each point, name the ais it is on or the quadrant it lies in. D C E F A G B Solution Points A and E lie on the -ais. Points F and G lie on the -ais. Point C lies in Quadrant II. Point D lies in Quadrant III. Point B lies in Quadrant IV. TIP Ever point in a coordinate plane lies either on an ais or in a quadrant, but not both. POINTS ON THE PLANE 0 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Using an Ordered Pair to Describe a Location To describe the location of a point, use an ordered pair. An ordered pair has the form (, ). The number (the -coordinate) describes the point s horizontal (left-right) distance from the origin. The number (the -coordinate) describes the point s vertical (up-down) distance from the origin. The numbers in an ordered pair are called coordinates. To name the coordinates of a point, determine how ou can get to the point from the origin b first counting right or left and then counting up or down. Eample Name the ordered pair for each point. N P M Q Solution A Point M N P M units units Q Start at the origin. Go units to the right. Go units up. THINK ABOUT IT Point M is directl above on the -ais and directl to the right of on the -ais. Therefore, its coordinates are (, ). The ordered pair for point M is (, ). B Point N N units P M Q The ordered pair for point N is (, 0). Start at the origin. Go units to the left. Go 0 units verticall (up or down). 0 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
C Point P N units units P M Q Start at the origin. Go units to the left. Go units down. REMEMBER The signs for the directions are + for right and up and for left and down. The ordered pair for point P is (, ). D Point Q N P M units units Q Start at the origin. Go units to the right. Go units down. The ordered pair for point Q is (, ). Graphing an Ordered Pair Eample Graph each ordered pair on the coordinate plane. Name the quadrant in which the point lies or the ais on which the point lies. A (, ) Solution REMEMBER A point is the graph of an ordered pair. An ordered pair contains the coordinates of a point. (, ) units unit Start at the origin. The -coordinate is. Go unit left. The -coordinate is. Go units up. Draw and label a dot. The point (, ) lies in Quadrant II. (continued) POINTS ON THE PLANE 0 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B (0, ) Solution units (0, ) The point (0, ) lies on the -ais. C (, ) Solution Start at the origin. The -coordinate is 0, so do not move left or right. The -coordinate is. Go units down. Draw and label a dot. THINK ABOUT IT Ever point on the -ais has -coordinate 0. Ever point on the -ais has -coordinate 0. units units (, ) Start at the origin. The -coordinate is. Go units right. The -coordinate is. Go units down. Draw and label a dot. The point (, ) lies in Quadrant IV. 0 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Problem Set Use the coordinate plane below for problems. A F H E B J C G D K For each point, A. Name the ais it is on or the quadrant it lies in. B. Name the ordered pair.. A. F. B 7. G. C. H. D 9. J. E. K Graph each ordered pair on a coordinate plane.. (, ). (, ). (0, ). (, 0). (, ). (, ) 7. (, ). (, ) 9. (, ) 0. (, ) Solve. *. Challenge Start at the origin. Go units right, units down, units left, units right, and units up. Which quadrant are ou in? POINTS ON THE PLANE 07 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Two-Variable Equations A solution of a one-variable equation is a single number. For eample, the solution of = is. But a solution of a two-variable equation is an ordered pair of numbers. Determining Whether an Ordered Pair Is a Solution An ordered pair is a solution to a two-variable equation in and if substituting the first coordinate for and the second coordinate for results in a true equation. Eample Determine if the ordered pair is a solution of the equation. A Is (, ) a solution of + =? Solution + = Write the two-variable equation. + Substitute for and for. + Multipl. = Add. Yes, (, ) is a solution of + =. B Is (, ) a solution of s = r? Solution s = r Write the two-variable equation. Substitute for r and for s. Multipl. Subtract. No, (, ) is not a solution of s = r. C Is (0., ) a solution of =? Solution = Write the two-variable equation. 0. ( ) Substitute 0. for and for. + Simplif. = Add. Yes, (0., ) is a solution of =. TIP Most equations with two variables have countless ordered-pair solutions. TIP In the equation s = r, the ordered pairs are traditionall written in the form (r, s) because r comes before s in the alphabet. 0 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Finding Solutions of an Equation In the equation = +, ou can choose an value for the variable. Once ou substitute a value for, ou can solve to find the corresponding value of. You could also start with a value of. Eample Find three solutions of each equation. A = Solution Choose an values for. Find the corresponding values of. Let =. Let = 0. Let =. = = = = ( ) = 0 = = = 0 = = = = (, ) is a solution. (0, ) is a solution. (, ) is a solution. Three solutions of = are (, ), (0, ), and (, ). B 9 = Solution Step Solve for. 9 = Write the equation. 9 = Divide both sides b. = Simplif. Step Choose values for and find the corresponding values for. Let =. Let =. Let =. = = = ( ) = = = + = = = 7 = = 7 = (, 7) is a solution. (, ) is a solution. (, 7) is a solution. Three solutions of 9 = are (, 7), (, ), and (, 7). Graphing an Equation The graphs of all the ordered-pair solutions of an equation form the graph of the equation. You can record several solutions in a table of values and then graph those ordered pairs. After graphing several ordered pairs, ou can use the pattern ou have formed to draw the graph of the equation. Eample Graph =. Solution First, solve the equation for. = Write the equation. + = + Add to both sides. = + Simplif. (continued) THINK ABOUT IT The number of possible choices for is unlimited, so the number of ordered-pair solutions is unlimited. TWO-VARIABLE EQUATIONS 09 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Net, record several ordered pair solutions in a table of values. + (, ) ( ) + = + = (, ) ( ) + = + = (, ) 0 0 + = 0 + = (0, ) + = + = 0 0 (, 0) + = + = (, ) TIP Choose values for that are eas to work with. Zero is usuall an eas value to work with. (, ) (, ) (0, ) (, 0) (, ) Graph the ordered pairs. The points lie in a line. Draw a line through the points. The line is the graph of =. THINK ABOUT IT An linear equation has an infinite number of solutions. Their corresponding points form a line, which has infinitel man points. Determining Dependent Variables and Independent Variables In a two-variable equation, the independent variable is the input variable and the dependent variable is the output variable. Sometimes ou need to determine which variable is dependent and which variable is independent. Consider these two variables: price p and sales ta t. It is not reasonable to sa that the price of an item depends on the amount of sales ta charged on that item. Instead, it is reasonable to sa that the amount of sales ta t depends on the price p. So in this case, the dependent variable is t and the independent variable is p. Eample Identif the dependent variable and the independent variable in each situation. Eplain our answer. A Mai is a sales associate. She earns a salar of $0 per week and a % commission on sales. If s is the amount of Mai s sales in dollars and p is Mai s total pa, then the variables s and p are related b the equation 0.0s + p = 0. Solution Because Mai s total pa p depends on the amount of her sales s, the dependent variable is p and the independent variable is s. TIP In an ordered pair, the independent variable is first and the dependent variable is second. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B Rico has gallons of gasoline in his car s tank when he begins a road trip. During the trip, g is the number of gallons remaining in the tank and m is the number of miles he has driven. The variables are related b the equation 0.0m + g =. Solution Because the number of remaining gallons g depends on the number of miles driven m, the dependent variable is g and the independent variable is m. C Tickets to a school pla cost $.0 for adults and $ for students. On opening night, $90 is collected in ticket sales. If a is the number of adult tickets sold and s is the number of student tickets sold, then the variables are related b the equation.0a + s = 90. Solution There is no wa to determine whether sales of one tpe of ticket affect sales of the other tpe of ticket. So in this case, it is not reasonable to sa that either variable depends on the other. If ou need to decide that one variable is dependent and the other is independent, then ou can decide either wa. Problem Set Determine if the ordered pair is a solution of the equation.. Is (0, 0) a solution of =?. Is (, ) a solution of =?. Is (, ) a solution of =?. Is (, ) a solution of + =?. Is (, ) a solution of + = 9?. Is (, ) a solution of + = 9? 7. Is (, ) a solution of = 0?. Is (0, 7) a solution of 9 + =? 9. Is (, 7) a solution of = +?. Is (, ) a solution of = +?. Is (, ) a solution of =?. Is (.,.) a solution of + = 0? Find three solutions of each equation.. =. q = p +. f = e. h = g 7. 0 =. = 9 * 9. Challenge 9 = * 0. Challenge = + TWO-VARIABLE EQUATIONS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Make a table of values, and then graph each equation.. =. =. =. + =. = *. Challenge = + * 7. Challenge = + Identif the dependent variable and the independent variable in each situation. Eplain our answer.. Mart borrowed mone from his mother to bu a new car. He makes a pament of $0 to her each month. The total amount a that Mart still owes his mother is given b the equation a = 000 0m where m is the number of monthl paments Mart has alread made. 9. An adventure fitness club charges a $ membership fee. A one-hour session of rock-wall climbing costs $. If n is the number of climbing sessions and t is the total amount spent in fees and climbing sessions combined, then the variables n and t are related b the equation t n =. 0. Caroline has n nickels and d dimes in her pocket. The total value of the coins is $.. The variables are related b the equation 0.0n + 0.d =... A prepaid cellular phone is loaded with a $0 card. Each minute of talk costs $0.. If r is the remaining balance on the card and t is the number of minutes talked, then the variables are related b the equation r + t = 00. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Linear Equations and Intercepts A line is made up of an infinite number of points and etends without end in two opposite directions. Identifing the Intercepts of a Line When ou graph a line on a coordinate plane, it crosses at least one ais. DEFINITIONS The -intercept of a line is the -coordinate of the point where the line intersects the -ais. The -intercept of a line is the -coordinate of the point where the line intersects the -ais. Eample Name the intercepts of each line. A Solution The line intersects the -ais at (, 0). The -intercept is. The line intersects the -ais at (0, ). The -intercept is. B Solution The line is horizontal. It does not intersect the -ais, so it has no -intercept. The line intersects the -ais at (0, ). The -intercept is. LINEAR EQUATIONS AND INTERCEPTS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Using Intercepts to Graph A + B = C If the graph of an equation is a line, the equation is a linear equation. The graph of an equation in the form A + B = C (where A, B, and C are integers and A and B are not both zero) is a line. An equation in this form is called the standard form of a linear equation. If A and B are both nonzero, the line is neither horizontal nor vertical, and it will intersect both aes. GRAPHING A LINEAR EQUATION BY FINDING ITS INTERCEPTS Step Find the -intercept b letting = 0 and solving for. Step Find the -intercept b letting = 0 and solving for. Step Graph the intercepts. Draw the line through both points. Eample Graph each equation. A + = Solution First, find the intercepts. + = Write the equation. + 0 = To find the -intercept, let = 0. = Simplif. = Solve for. The -intercept is. The line intersects the -ais at (, 0). Net, find the -intercept. + = Write the equation. 0 + = To find the -intercept, let = 0. = Simplif. = Solve for. The -intercept is. The line intersects the -ais at (0, ). (0, ) (, 0) Graph the intercepts. Draw the line that contains both points. The line is the graph of + =. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B = Solution First, write the equation in standard form. = Write the equation. + = + Add to both sides. = + Simplif. = + Subtract from both sides. = The equation is in standard form. Net, find the intercepts. = = 0 = 0 = = = = = The -intercept is. The -intercept is. REMEMBER Equations that have all the same solutions are equivalent. In Eample B, the equations = and = are equivalent. (, 0) (0, ) Graph the intercepts. Draw the line through both points. The line is the graph of =. Graphing = b The graph of an equation in the form = b, where b is a constant, is a horizontal line with -intercept b. Eample Graph =. Solution The graph is the horizontal line with -intercept. (0, ) Graph the -intercept. Draw a horizontal line through the point. The line is the graph of =. THINK ABOUT IT Some solutions to the equation = are (0, ), (, ), and (, ). The -coordinate of ever point on the line is. LINEAR EQUATIONS AND INTERCEPTS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Graphing = a The graph of an equation in the form = a, where a is a constant, is a vertical line with -intercept a. Eample Graph =. Solution The graph is the vertical line with -intercept. REMEMBER The graph of = a is a vertical line. The graph of = b is a horizontal line. (, 0) Graph the -intercept. Draw a vertical line through the point. The line is the graph of =. Using Intercepts to Write an Equation of a Line USING INTERCEPTS TO WRITE AN EQUATION OF A LINE Step Let C equal the product of the intercepts. Step Substitute the ordered pairs that contain the intercepts into A + B = C to find the values of A and B. Step Substitute the values of A, B, and C into A + B = C. Eample Write a linear equation for the graph that is shown. (, 0) (0, ) Solution The intercepts are and. The product is =. Use (0, ) and (, 0) to find the values of A and B. Substitute (0, ) Substitute (, 0) A + B = A + B = A 0 + B = A ( ) + B 0 = B = A = B = A = Since A =, B =, and C =, the equation is =. THINK ABOUT IT You could use an nonzero value for C, but using the product of the intercepts makes for neater numbers. TIP Check our equation b finding the intercepts. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Problem Set Name the intercepts of each line....... Find the intercepts of the graph of the line. 7. + =. = 9. + = 9. =. 9 + =. = LINEAR EQUATIONS AND INTERCEPTS 7 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Graph each equation.. =. + =. = 7. = 7. =. = 9. = 0. =. + =. =. =. + = 9 *. Challenge + = + *. Challenge + = Write a linear equation for the graph that is shown. 7. 0. (, 0) (7, 0) (0, ).. (, 0) (0, ) (0, ) 9.. (0, ) (, 0) (, 0) (0, ) UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Slope Lines can var in steepness. The can also var in whether the slant up to the left, up to the right, or not at all. Slope describes all those variations. Calculating Slope Let (, ) and (, ) be two points on a line. When moving from (, ) to (, ) on the line, the vertical change, or rise, equals and the horizontal change, or run, equals. SLOPE The slope of a line is the ratio of the rise to the run. slope = rise run = vertical change horizontal change = TIP is read sub, and means the -coordinate of the first point. Eample Find the slope of the line. (, ) (, ) Solution You can count the rise and the run from one point to another. Or, ou can use the slope formula. Let (, ) be (, ) and let (, ) be (, ). Substitute the values of,,, and into the formula. (continued) SLOPE 9 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
rise: units (, ) (, ) The slope is. run: units (, ) (, ) slope = rise run = = ( ) ( ) = + + = = Calculating Slope More Than One Wa To find the slope of a line, use an pair of points on the line. And for an pair of points, ou can use either order. Eample Find the slope of the line. Use two different pairs of points and change the order for one pair of points. 9 7 (, ) (, ) (, 0) 7 9 Solution (, ) (, ) slope = (, ) (, ) slope = = = (, 0) (, ) slope = 0 = = (, ) (, 0) slope = 0 = = This is the same pair of points in a different order. The slope is. 0 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Calculating the Slope of a Horizontal Line Eample Find the slope of the horizontal line. (, ) (, ) Solution slope = = = 0 (, ) (, ) = 0 The slope is 0. REMEMBER An fraction with a numerator of zero and a nonzero denominator equals zero. 0 a = 0 if a 0 Describing the Slope of a Vertical Line Eample Find the slope of the vertical line. Solution slope = = = 0 (, ) (, ) (, ) (, ) The slope is undefined. REMEMBER An fraction with denominator zero is undefined. a is undefined for all values of a. 0 SLOPE Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Using Slope to Classif and Describe Lines CLASSIFYING LINES BY SLOPE Positive Slope Negative Slope Zero Slope Undefined Slope The line rises The line falls The line is The line is left to right. left to right. horizontal. vertical. Eample Find the slope of the line that passes through the points (, 0) and (, ). Describe the line. Solution Let (, 0) be (, ) and let (, ) be (, ). slope = = 0 = = The line falls from left to right because it has a negative slope. TIP You can let either point be (, ). Then let the other point be (, ). Using Slope to Compare the Steepness of Lines The slope is a measure of the steepness of a line, that is, the angle the line makes with the horizontal. The steeper of two lines has the greater absolute value of slope. Eample Which is the slope of the steeper line, or? Solution Compare the absolute values of the slopes. = = and =. Since >, is the slope of the steeper line. THINK ABOUT IT When ou think about the steepness of a line, think about riding a bike or skateboard downhill on a street with that slope. The steeper the slope (greater absolute value), the faster ou will go. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Finding the Slope of a Line, Given the Equation in Standard Form You can use the intercepts as two points to find the slope of a line. Eample 7 Find the slope of the graph of =. Solution Find the -intercept. Find the -intercept. = = 0 = 0 = = = = = -intercept: (, 0) -intercept: (0, ) Graph the line. Find the rise and run. rise: units run: units (0, ) 7 (, 0) Finall, calculate the slope. slope = rise run = = SLOPE OF THE GRAPH OF A + B = C The slope of the graph of a linear equation in standard form A + B = C is A B. Eample Find the slope of the graph of =. Solution Identif the values of A and B, and then find the slope. A + B = C = A = and B =. slope = B A = = THINK ABOUT IT Eamples 7 and have the same answer because the have the same equation. Eample illustrates the slope propert for a linear equation in standard form. SLOPE Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Problem Set Use the graph to find the slope of each line... (, ) (, ) (, ) (, ).. (, ) (, ) (, ) (, ).. (, ) (, ) (, ) (, ) Find the slope of the line that passes through the points. 7. (, ) and (, ). (0, ) and (, 7) 9. (0, 0) and (, ). (, ) and ( 7, ). (, 9) and (, ). (, ) and (, ). (, ) and (, 7) *. Challenge (, ) and (, ) UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Find the slope of the line with the given equation.. + =. + = 7. =. 9 = 9. = 0. = *. Challenge = 7 *. Challenge + = 7 State whether the slope of the line is positive, negative, zero, or undefined.... 7... Which is the slope of the steepest line? 9. or 0.,, or. or SLOPE Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Applications: Linear Graphs Man applications can be modeled with linear equations. When data form a linear pattern, ou can use a linear graph to make inferences and predictions. TIP When ou infer, ou make an educated guess. Application: Health DEFINITION Interpolation is a process of inferring, or estimating, an unknown value that is between known values. THINK ABOUT IT Interpolate comes from the Latin inter meaning between and polus meaning point. Eample Mr. Nelson has been losing weight steadil for months. The graph shows data about his weight over time. What is a reasonable guess for Mr. Nelson s weight in month 7? Weight (lb) 0 0 00 Mr. Nelson s Weight 0 7 Month 9 Solution The data points form a linear pattern. Draw the line through the points. Then interpolate to infer the weight for month 7. Weight (lb) 0 0 00 Mr. Nelson s Weight 0 7 Month 9 It is reasonable to infer that Mr. Nelson s weight in month 7 was pounds. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Application: Personal Finance DEFINITION Etrapolation is a process of inferring, estimating, or predicting a value that is outside of known values. Eample Sheila has been saving mone at a stead rate. She is saving for a new television that costs $00. A graph of her savings is shown below. What is a reasonable prediction of when Sheila will have enough mone saved to bu the television set? THINK ABOUT IT Etrapolate comes from the Latin etra meaning outside and polus meaning point. When ou etrapolate, ou make an inference about values that are outside the known points. Month Total Saved ($) 00 00 900 0 Total saved ($) 000 700 00 0 0 00 0 900 00 00 Sheila s Savings 0 0 Month 00 7 9 Solution The data points form a linear pattern. Draw the line through the points. Then etrapolate to predict the first month in which Sheila s total savings will be at least $00. Month Total Saved ($) 00 00 900 0 Total saved ($) 000 700 00 0 0 00 0 900 00 00 Sheila s Savings 0 0 Month 00 7 9 Sheila should reach her savings goal in Month. The prediction is reasonable because Sheila is saving $00 per month. If she continues saving at the same rate, she will have $00 saved in month. APPLICATIONS: LINEAR GRAPHS 7 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Application: Sports In some cases, using etrapolation will not ield a reasonable prediction. Eample The number of runs scored b a baseball team in each of its first four games of a season are shown in the graph below. Use etrapolation to predict the number of runs the team will score in the 0th game of the season. Is the prediction reasonable? Eplain. Score Game Score Score 0 0 0 0 Game Solution The data points form a linear pattern. Draw the line through the points. Then etrapolate to predict the number of runs the team will score in the 0th game. Score Game Score Score 0 0 0 0 Game If the pattern continues, the team will score runs in the 0th game. This prediction is not reasonable because it is not reasonable to epect the pattern to continue. A constant increase of one run per game is not likel over more than a few games. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Problem Set Solve. For each set of problems use the given graph and information. Use the following information to solve problems. Because of a drought, the water level in a lake has been decreasing steadil for the past months. The graph shows data about the water level over time. Use the following information to solve problems. Washington Elementar School has eperienced an increase in enrollment each ear since it opened. The graph shows data about the student population over time. Lake Water Level 900 Washington Elementar 00 700 Water level (ft) 0 0 0 0 Student population 00 00 00 00 00 0 0 0 0 Week 7 9 0 0 7 Year 9. What is a reasonable inference for the water level in Week?. What is a reasonable prediction for the water level in Week 9?. What is a reasonable prediction for the water level in Week?. A state of emergenc will be declared when the water level in the lake reaches 0 feet. Use etrapolation to predict when a state of emergenc will be declared.. Is the prediction ou made in problem reasonable? Eplain.. What is a reasonable inference for the student population in Year? 7. What is a reasonable inference for the student population in Year 7?. What is a reasonable prediction for the student population in Year? 9. School administrators plan to build a new elementar school when the student population at Washington Elementar reaches 700. Use etrapolation to predict the ear in which that might happen.. Is the prediction ou made in problem 9 reasonable? Eplain. APPLICATIONS: LINEAR GRAPHS 9 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Use the following information for problems. A new car loses value as it ages. The value of a certain car during its first five ears is shown in the graph. Use the following information for problems 0. Marvin is lifting weights to build muscle. During his first four weeks of weight training, he tracks the greatest weight he can bench press each week in a graph. Car value ($) 000 0000 000 000 000 Car Value Greatest weight pressed 0 0 00 0 0 0 0 00 0 Bench Press 0 0 7 Year 9 0 Week 7 9. What is a reasonable inference for the value of the car in Year?. What is a reasonable prediction for the value of the car in Year?. If ou etrapolate from the data in the graph, what will be the value of the car in Year 9?. Use etrapolation to predict the ear in which the car will be worth nothing.. Is the prediction ou made in problem reasonable? Eplain.. During the first four weeks, how much does Marvin s heaviest press increase each week? 7. What is a reasonable prediction for Marvin s heaviest press in Week?. What is a reasonable prediction for Marvin s heaviest press in Week? 9. Use etrapolation to predict Marvin s heaviest press in Week 0. 0. Is the prediction ou made in problem 9 reasonable? Eplain. 0 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Relations and Functions A relation describes how two sets are related to each other. Identifing Domain and Range DEFINITIONS A relation is a correspondence between two sets. A relation can be shown with a table or a set of ordered pairs or a graph. The domain of the relation is the set of all first elements of the ordered pairs, called inputs. The range is the set of all second elements of the ordered pairs, called outputs. Eample Identif the domain and range. A {(, ), (, ), (, 0), (, 7), (, 7)} Solution Identif the first elements. Identif the second elements. Domain: {,,,, } Range: {,, 0, 7} B 0 TIP Braces { } are used to indicate a set. A relation is a set that contains ordered pairs. The domain and range of a relation are sets that contain single numbers. Solution The domain is the set of -values. The range is the set of -values. Domain: {, 0, } Range: {,,, } Determining Whether a Relation Is a Function DEFINITION A function is a relation that assigns eactl one output to ever input. That means that for ever ordered pair (, ) in a function, each value of is assigned eactl one value of. TIP A function is like a machine. When ou put in an allowable number, ou get a specific answer. (continued) RELATIONS AND FUNCTIONS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Eample Determine whether each relation is a function. A {(, ), (, 0), (, ), (, ), (, 7)} Solution Look at the ordered pairs (, 0) and (, ). The input is assigned two different outputs: 0 and. This relation is not a function. B 0 C 0 TIP If an -value is repeated, appearing in different ordered pairs, then the relation is not a function. Solution The input is assigned Solution Each input is assigned two different outputs: 0 and. This eactl one output. This relation is relation is not a function. a function. Appling the Vertical Line Test All relations can be graphed. When a relation is graphed, use the vertical line test to determine whether it is a function. VERTICAL LINE TEST If an vertical line can be drawn to intersect the graph of a relation in more than one point, the relation is not a function. If no vertical line eists that intersects the graph in more than one point, the relation is a function. THINK ABOUT IT If a vertical line hits more than one location on a graph, it means there are two points with the same -coordinate. Eample Appl the vertical line test to determine whether each graph represents a function. A Solution A vertical line intersects the graph in more than one point. The graph fails the vertical line test. The graph does not represent a function. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B Solution There is no vertical line that intersects the graph in more than one point. The graph passes the vertical line test. The graph does represent a function. C Solution A vertical line intersects the graph in more than one point. The graph fails the vertical line test. The graph does not represent a function. The vertical line test is reall just a wa of appling the definition of a function. Note that in the preceding Eample A, the points on the vertical line are (, ) and (, ). Therefore, the input is assigned two different outputs: and. B definition, the relation is not a function. TIP The arrowheads on the graph in Eample B indicate that the graph continues without end. Evaluating Functions Man functions can be represented b equations. For instance, the equation = + represents a function. Two solutions of the equation are (, ) and (, ). All the solutions of the equation form the set of ordered pairs that make up the function. If a function equation has variables and, then is the input (independent) variable, and is the output (dependent) variable. We sometimes use the notation f () in place of the variable. In function notation, is the input variable. Using function notation, the equation = + can be written f () = +. If =, then = f () = f () = + =. This is called evaluating the function f for the input =. Note that the values = and = are in the ordered pair (, ). THINK ABOUT IT Read f() as f of. TIP Variables other than,, and f can be used for functions. For eample, the function d = 0t gives distance d as a function of time t. It can be written d(t) = 0t. (continued) RELATIONS AND FUNCTIONS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Eample Evaluate the function for the given input. A Find f (7). Solution f() 0 7 7 f() 0 7 7 Find the input 7 in the table. The output is. f (7) = B Find f ( ). Solution (, ) Find the -coordinate when the -coordinate is. f ( ) = C For f () =, find f (). Solution f () = Write the function equation. f () = Substitute for. = Multipl. = Subtract. f () = Problem Set Identif the domain and range.. {(, 0), (, ), (, 0), (0, ), (, )}. {(, ), (7, ), (, ), (, ), (, )}. {(, ), (, ), (, ), (, ), (, )} UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
... 0 0 0 0 0 Determine whether each relation is a function. 7. {(, 0), (, 7), (, ), (, ), (9, )}. {(0, ), (, ), (, ), (, ), (, )} 9. {(, ), (, ), (, ), (, ), (, )}... 0 0 0 Appl the vertical line test to determine whether each graph represents a function..... RELATIONS AND FUNCTIONS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Evaluate the function for the given input. 7. f () f() 0 7 9. g(9) g() 9 0 9. h( ) h() 0. f ( ) f(). s() s(). f () f(). g(9) g(). f () =. g() =. f () = + 7. h() = + Find f (). Find g( ). Find f (). Find h(0). UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Sstems of Linear Equations A sstem of linear equations is made up of two or more linear equations. A solution of a sstem of linear equations in two variables is an ordered pair that is a solution of each equation in the sstem. Solving a Sstem of Linear Equations b Graphing Eample Solve the sstem b graphing. = = + Solution Graph each equation. The ordered-pair coordinates of the point of intersection is the solution. = 0 7 = + 9 0 0 Make a table of values for each equation. TIP Find at least ordered pairs for each line. Choose -values that are spaced apart. This helps ensure that ou draw an accurate line. = + = (, ) Graph each equation. Identif the point of intersection. THINK ABOUT IT Each equation in the sstem has an infinite number of orderedpair solutions, corresponding to the infinite number of points on each line. But onl the point (, ) is on both lines. So, (, ) is the onl solution of the sstem. The point of intersection appears to be at (, ). (continued) SYSTEMS OF LINEAR EQUATIONS 7 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Check Substitute (, ) into each equation. The solution is correct if each equation is true. = = + + = = Both equations are true for (, ). The solution of the sstem is (, ). Determining the Number of Solutions of a Sstem of Linear Equations A sstem of linear equations in two variables can have eactl one solution, no solutions, or an infinite number of solutions. Eample Use the graph to determine the number of solutions of the sstem of equations. A = + + = + = = + Solution The lines are parallel. There is no point of intersection. The sstem has no solution. UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B = =. = =. Solution The two equations in the sstem describe the same line. Ever point on the line is a solution to each equation. The sstem has an infinite number of solutions. C = + = = + = THINK ABOUT IT The lines in the graph of a linear sstem can be slanted, vertical, or horizontal. Solution The lines intersect in one point. There is eactl one ordered pair that is a solution of both equations in the sstem. The sstem has eactl one solution. Solving a Sstem of Linear Equations b Substitution You can solve a sstem of linear equations b algebraic methods. One algebraic method is substitution. SOLVING A SYSTEM OF LINEAR EQUATIONS BY SUBSTITUTION Step Step Step Solve one of the equations for one of the variables. Substitute the epression from Step into the other equation and solve for the other variable. Substitute the value ou found in Step into either equation and solve for the remaining variable. (continued) SYSTEMS OF LINEAR EQUATIONS 9 Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Eample A Solve the sstem b substitution. + = 0 Solution = + Step The equation = + is alread solved for. Step Substitute + for into the other equation and solve for. + = 0 Write the first equation. + + = 0 Substitute + for. + = 0 = Simplif the left side of the equation. Subtract from both sides. = Divide both sides b. Step Substitute for into either equation. = + Write the second equation. = + Substitute for. = Simplif to solve the equation. The result in Step is =. The result in Step is =. The solution of the sstem is (, ). B Solve the sstem b substitution. + = 9 + = Solution Step Solve the first equation for. + = 9 = 9 Step Substitute 9 for into the second equation and solve for. + = Write the second equation. 9 + = Substitute 9 for. 9 + = Simplif the left side of the equation. = Subtract 9 from both sides. Step Substitute for into either equation. + = 9 Write the first equation. + = 9 Substitute for. + = 9 Simplif the left side of the equation. = Subtract from both sides. The solution of the sstem is (, ). THINK ABOUT IT For Step, ou could solve + = 0 for. Then for Step, ou would substitute that epression into = +. TIP Check the solution b substituting (, ) into both equations in the sstem. 0 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Application: Ticket Sales Eample Tickets to a school pla cost $ for adults and $ for students. On opening night, tickets were sold and $70 were collected in ticket sales. How man of each tpe of ticket were sold? Solution Let a be the number of adult tickets sold and s be the number of student tickets sold. Write and solve a sstem of equations. a + s = a + s = 70 Solve the first equation for s. s = a A total of tickets were sold. a + s dollars were collected from ticket sales. Subtract a from both sides. Substitute this epression for s in the second equation and solve for a. a + ( a) = 70 a + 0 a = 70 a + 0 = 70 a = 90 Substitute a for s into the second equation. Distribute. Simplif the left side of the equation. Subtract 0 from both sides. s = a = 90 = 0 Substitute 90 for a to find the value of s. a = 90 and s = 0. There were 90 adult tickets and 0 student tickets sold. THINK ABOUT IT Eample is a case in which it is not reasonable to sa that either variable depends on the other. You can choose either variable to be the independent variable. If ou choose a to be the independent variable, then ordered pairs are in the form (a, s), and the solution of the sstem is (90, 0). Problem Set Use the graph to determine the number of solutions of the sstem of equations... = = + = = SYSTEMS OF LINEAR EQUATIONS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
.. = 0. = = =.. = = = = Solve each sstem b graphing. 7. = =. = = 9. = =. = =. = =. + = =. + = + =. + = 9 = UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Solve each sstem b substitution.. = + + =. = = 7. = = +. = + = 9. = = 0. = + = * *. + = + =. + = =. = + =. = + =. Challenge + = 9 =. Challenge + = 0 = Write and solve a sstem of equations to answer each question. 7. Mrs. Young teaches flute and piano lessons. She has 9 flute students. The total number of students is. How man piano students does Mrs. Young have?. The sum of two numbers is. The difference of the numbers is 7. What are the two numbers? 9. B weight, a trail mi contains times as much raisins as peanuts. The total weight of the peanuts and raisins in the miture is ounces. How man ounces of peanuts are in the miture? How man ounces of raisins? * 0. Challenge Charlie bought large pizzas and drinks and paid $. At the same restaurant, Allen bought large pizzas and drinks and paid $7. What is the cost of a large pizza? What is the cost of a drink? SYSTEMS OF LINEAR EQUATIONS Copright 00, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.