Section. Graphing Lines Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif collinear points. The order of operations (1.) Graph the line of an equation of the Placing values into formulas (1.8) form a b = c. Plotting points in the --plane (.1) Graph the line of an equation of the Drawing a line in the --plane (.1) form = m b. Graph a line using a different scale. INTRODUCTION A linear equation of two variables, and, is one in which the ordered pair solutions are points on a line in the --plane. Because the ordered pairs are both solutions and points, there will be times in this tet when the are both referred to as points. In Section.1 we saw a few linear equations, including =, = 1, and = -. Each of these equations was graphed as a line in the --plane, and even though the fewest number of points needed to draw a line is two, we usuall plot at least three points to help us draw the line more accuratel. COLLINEAR POINTS Here, again, are some of the solutions to the linear equation =. There are an infinite number of solutions, each an ordered pair, and those ordered pairs can be plotted in the --plane. In table form As ordered pairs As points in the --plane (, ) 7 (7, ) 5.5 1.5 (5.5, 1.5) 0 (, 0) 1 - (1, -) and man more... (7, ) (, 0) (5.5, 1.5) -8 - - - 8 - Two or more points that form a line or are alread on the same line are said to be collinear (co-lih nee-er). - - (1, -) Graphing Lines Robert Prior, 010 page. - 1
Because these ordered pairs came from the same linear equation, =, the corresponding points in the --plane are collinear and we can draw a (straight) line through them. As the line etends through the --plane, we can see other points on the line, such as (0, -) and (-, -). (7, ) (, 0) (5.5, 1.5) -8 - - - 8 - Do we need all of these points to draw the line? No. In fact, as stated in Section.1, we can use as few as two points, because an two points determine a unique line. - - (1, -) Do an three points determine a unique line? No. Three points determine a unique line onl if the points are collinear. In fact, if three points are not collinear, then the actuall determine three lines, as demonstrated in Eample 1. Eample 1: Procedure: For this set of three non-collinear points, draw the three lines that pass through each pair. (, 5), (-, ) and (-, -). Plot the points and draw the three lines connecting two of them at a time. Answer: (-, ) (, 5) (-, ) (, 5) -8 - - - 8 - -8 - - - 8 - (-, -) - (-, -) - - - Graphing Lines Robert Prior, 010 page. -
YTI 1 For each set of three non-collinear points, draw the three lines that pass through each pair. Use Eample 1 as a guide. a) (-, ), (, ), and (, -) b) (-, 0), (0, -5), and (, ) -8 - - - 8 - -8 - - - 8 - - - - - LINEAR EQUATIONS OF THE FORM a b = c In Section.1 ou were introduced to the equation =. This is a member of a famil of linear equations of the form a b = c, where a, b, and c are numbers and and are variables. Linear equations with two variables, and, can be written in this form: a b = c Here, a, b, and c are numbers and and are variables. ~Instructor Insight~ The standard form of an equation is discussed in detail in Section.. The equation A B = C will be referred to as the standard form equation at that time. ~Instructor Insight~ The graphing of horizontal and vertical lines is not discussed in this section. Instead, the are discussed in Section.. Graphing Lines Robert Prior, 010 page. -
We can graph an line that has this form b finding three ordered pair solutions, and we often have a choice as to which three points are to be plotted. However, not all points on a line are eas to graph. For eample, a point such as (1, 17) would not normall fit in our --plane. We can find the coordinates of a point b either: 1. choosing a value of, placing it into the equation, and then solving for, or. choosing a value of, placing it into the equation, and then solving for. Often, one point can be found b choosing = 0, and another b choosing = 0. Then, b choosing one other value of either or, we will have three points to plot and can draw the line associated with the linear equation. ~Instructor Insight~ The - and -intercepts are not introduced here. The are discussed in detail in Sections. and.. Eample : Procedure: Graph the line of the equation = - b first finding three ordered pair solutions. For this eample, let s choose = 0, = 0, and = 1. We ll set up a table to keep our work organized. (Be careful to place the - and -coordinates correctl in the ordered pair.) Answer: Choose a value = - (, ) = 0 = 0 (0) = - 0 = - = - 0 = - = - = - = - (0, - ) (-, 0) (-, 0) -8 - - - 8 - - - (1, -5) = 1 (1) = - = - (- ) = - (- ) = - 5 (1, - 5) Graphing Lines Robert Prior, 010 page. -
Caution: If the three points we plot are not on the same line, then we made a mistake somewhere, and we must correct it before we can graph the line. YTI Graph the line of the equation = b first finding three sets of ordered pair solutions. Use these values: = 0, = 0, and =. Use Eample as a guide. Choose a value = (, ) = 0 = 0 = -8 - - - 8 - - - YTI Graph the line of the equation = b first finding three sets of ordered pair solutions. You choose the three values to use. Use Eample as a guide. Choose a value = (, ) -8 - - - 8 - - - Graphing Lines Robert Prior, 010 page. - 5
LINEAR EQUATIONS OF THE FORM = m b Another form of a linear equation is = m b. Written in this form, we sa that is in terms of. Because is alread isolated on the left side, we can choose three values of to find the three points. Linear equations written with in terms of have the form = m b Here, m and b are numbers and and are variables. ~Instructor Insight The slope and the -intercept are discussed in detail in Section.. The equation = m b will be referred to then as the slope-intercept form. Eample : Procedure: Graph the line of the equation = 1 b first finding three ordered pair solutions. For this eample, let s choose = 0,, and -. We ll set up a table to keep our work organized. Answer: = 1 (, ) 0 = (0) 1 = 0 1 = 1 = () 1 = 1 = 5 (0, 1) (, 5) -8 - - - 8 - (-, -) - (, 5) (0, 1) - = (- ) 1 = - 1 = - (-, -) - Graphing Lines Robert Prior, 010 page. -
We sometimes must be careful with the values of we choose. For eample, in the equation = 1, choosing = 10 gives = 1. The ordered pair (10, 1) is a solution, and there is such a point that can be plotted, but this would not fit in a tpical --plane. Instead, choose relativel small values of, somewhat close to 0. In fact, a good value to choose for is 0 itself. YTI Graph the line of each given equation b first finding three sets of ordered pairs solutions. Use Eample as a guide. a) = - b) = 1 a) = - (, ) b) = 1 (, ) -8 - - - 8 - -8 - - - 8 - - - - - Graphing Lines Robert Prior, 010 page. - 7
Think about it 1 Consider the equation = 1 5. What values of would ou choose to find four points on the line? Eplain our answer. A FRACTIONAL COEFFICIENT OF. Some linear equations of the form = m b have a fractional coefficient of. In such cases, it s best to choose values of that are multiples of the denominator. Doing so will tpicall result in integer values for. For eample, consider the equation =. Because the denominator is, we should choose values of that are multiples of, such as, -,, - and 0. As has been mentioned, we need choose onl three of these -values, so we might choose 0,, and, or mabe -, 0, and. Let s see. Eample : Procedure: For the equation =, find three sets of ordered pairs as points in the - plane. Draw the line that passes through these points. Choose three values of and find the corresponding values. In this case, choose values of that are multiples of. Answer: = (, ) 0 = 0 = 0 = = = = = = = (0, ) (, ) (, ) (, ) (, ) (0, ) -8 - - - 8 - - - Graphing Lines Robert Prior, 010 page. - 8
Notice the graph (and the equation) in Eample 8. We can see that there are two more points on the line that we can easil identif: (-, 0) and (-, -). Each of these points has an -values that is a multiple of. What would happen if we had chosen a value of that is not a multiple of? Would we still get a point on the line? Let s eplore this question b choosing = 1 and = -: = (, ) = (, ) 1 = (1) = = ( 1, ) = = 8 - = (-) = - = - (-, ) = = Both points ( 1, ) and ( ) line, as we can see on the graph. -, are on the These points are a bit more challenging to plot in the --plane because of the fractions in the - coordinates. (1, ) (-, ) -8 - - - 8 - - - (0, ) (, ) (, ) So, we can choose an value of and find the corresponding value of, but it is best to choose values of that are multiples of the denominator. Graphing Lines Robert Prior, 010 page. - 9
Eample 5: Procedure: For the equation = - 5 1, find three sets of ordered pairs as points in the - plane. Draw the line that passes through these points. Choose three values of and find the corresponding values. In this case, choose values of that are multiples of. Answer: = 1 (, ) 0-5 -5 = 0 1 = 0 1 = -1 (0, -1) (-, ) - -5 = 1 = -5 1 = - -5 = (-) 1 = 5 1 = (, -) (-, ) -8 - - - (0, -1) 8 - - - (, -) YTI 5 For each equation, find three sets of ordered pairs as points in the - plane. Draw the line that passes through these points. Use Eamples and 5 as guides. a) = 5 b) = - 1 5 a) = 5 (, ) b) = - 1 5 (, ) Graphing Lines Robert Prior, 010 page. - 10
-8 - - - 8 - -8 - - - 8 - - - - - YTI For each equation, find three sets of ordered pairs as points in the - plane. Draw the line that passes through these points. Use Eamples and 5 as guides. a) = - 1 b) = a) = - 1 (, ) b) = (, ) -8 - - - 8 - -8 - - - 8 - - - - - Graphing Lines Robert Prior, 010 page. - 11
GRAPHING LINES WITH A LARGE SCALE Sometimes a graph will have either - or -coordinates that do not fit easil on our tpical --plane. Consider, for eample, the equation = 10 0. If we choose values of such as -1, 0, and 1, as shown at right, then all of the points found will be above or below our tpical graph. We can still graph the line, but we must create an --plane that has a larger scale on the -ais. To create a larger scale, we make each grid line 5 or 10 (or more) times the normal -value. For this eample, let s see it done with two different scales. = 10 0-1 = 10(- 1) 0 = - 10 0 = 10 0 = 10(0) 0 = 0 0 = 0 1 = 10(1) 0 = 10 0 = 0 (, ) (- 1, 10) (0, 0) (1, 0) 5 times the normal -value scale 10 times the normal -value scale 0 (1, 0) 0 0 (0, 0) (-1, 10) - 8 - - - 8-10 (-1, 10) 0 0 (1, 0) (0, 0) - 8 - - - 8-0 - 0-0 - 0-0 These two lines represent the same line, = 10 0, even though the appear to have a different slant to them. The different slant is due to the different scales being used. Graphing Lines Robert Prior, 010 page. - 1
YTI 7 Graph the line of the given equation b first finding three ordered pairs solutions. Use the values of given in the table. a) = 10 5 b) = -0 0 a) = 10 5 (, ) b) = -0 0 (, ) - -1 0 0 1 5 times normal -value scale 10 times normal -value scale 0 0 0 0 10-8 - - - 8-10 0-8 - - - 8-0 - 0-0 - 0-0 Graphing Lines Robert Prior, 010 page. - 1
Answers: You Tr It and Think About It Some of the points ou find ma be different from the ones displaed in this answer set, but the should be on the same line. YTI 1: b) (, ) (-, ) -8 - - - 8 - (, -) - - (-, 0) (, ) -8 - - - 8 - - (0, -5) - YTI : YTI : Some points shown here ma be different from ours. (, ) - 8 - - - (, 0) 8 - (-, 5) (0, ) (-, ) (, ) (, 0) - 8 - - - 8 - - (0, - ) - - - Graphing Lines Robert Prior, 010 page. - 1
YTI : a) Some points shown here ma be different from ours. b) (-1, -5) (, 5) (0, ) -8 - - - 8-8 - - - (1, 0) 8 - (, -1) - - (-1, -) - - - YTI 5: Some points shown here ma be different from ours. a) (5, 0) -8 - - - 8 - (0, ) - b) (0, 5) (-, ) (, ) -8 - - - 8 - - (-5, -) - - YTI : Some points shown here ma be different from ours. a) (-1, -5) (0, ) (, ) -8 - - - 8 - b) (, ) -8 - - - (, -1) 8 - - - (0, -) - - Graphing Lines Robert Prior, 010 page. - 15
YTI 7: a) 5 times the normal -value scale b) 10 times the normal -value scale 0 0 (, 5) 10 (, 15) - 8 - - - 8-10 (0, -5) (- 1, 0) 0 0 (0, 0) - 8 - - - 8-0 (1, - 0) (-, -5) - 0-0 - 0-0 (, - 0) Think About It: 1. Answers ma var. One possibilit is, Good choices for are multiples of, such as 0,, -,, and -. Each of these values, when placed in the equation for, help to eliminate the fraction in the equation. Think Again. Section. Eercises 1. Consider the equation = 5. What values of would ou choose to find four points on the line? Eplain our answer. (Think about it #1). What is another name for the graph of = 0?. Three points in the --plane are not on the same line. How man lines can be drawn in the plane that pass through at least two of the points? Eplain our answer or show an eample that supports our answer. You ma want to draw a graph to assist ou in our eplanation.. Wh is it good to find at least three points on a line before graphing the line? Focus Eercises. Graph the line with the given equation b first finding three points on the line. 5. =. = - 1 7. = - 8. = Graphing Lines Robert Prior, 010 page. - 1
9. = 0 10. = 0 11. = 1. = 1. = - 1. = - 15. = - 1. = - 17. = 18. = - 19. = 0 0. 5 = 0 Graph the line with the given equation b first finding three points on the line. 1. =. = 5. =. = 5. = -. = - 1 7. = - 8. = - 9. = 0. = 1 1. = -. = -. =. = 5 5. = -. = - 7. = 1 5 8. = 1 9. = - 1 0. = - 1 5 1. =. =. = 1. = - 1 Graph the line with the given equation b first finding three points on the line. For each, a suggested larger scale is given. 5. = 10 15. = 0 0 Use a scale that is 5 times the normal scale. Use a scale that is 10 times the normal scale. 7. = - 0 0 8. = - 150 50 Use a scale that is 0 times the normal scale. Use a scale that is 50 times the normal scale. Think Outside the Bo. Plot each pair of lines (Line A and Line B) in the same --plane. Identif the point where the cross each other. Verif that the point found is a solution of each equation. 9. Line A: = 50. Line A: = - 1 Line B: = - Line B: = Graphing Lines Robert Prior, 010 page. - 17
The graph of the equations, below, are not lines. Instead the points are connected b smooth curves. The graphs are called parabolas. For each, find the ordered pairs and plot them in the --plane. Then connect the dots as smoothl as possible to form a parabola. 51. = 5. = = (, ) = (, ) -1-0 - 1 - -1 0 1 5 8 8-8 - - - 8 - - 8 - - - 8 - - - - - Graphing Lines Robert Prior, 010 page. - 18