10 GRAPHING LINEAR EQUATIONS
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1 0 GRAPHING LINEAR EQUATIONS We now expand our discussion of the single-variable equation to the linear equation in two variables, x and y. Some examples of linear equations are x+ y = 0, y = 3 x, x= 4, and y =. The graph of a linear equation is a line. Although drawing a line is a simple geometric construction, graphing a line requires knowledge of algebraic concepts The Cartesian Plane The Cartesian plane is a two-dimensional space formed by intersecting, in a perpendicular fashion, two number lines at their zeros. The number line placed horizontally is called the x-axis or abscissa and the number line placed vertically is called the y-axis or ordinate of the Cartesian plane. Since the number lines are of infinite length the resulting plane is of infinite area. Consequently, we can only draw the portion of the plane immediately surrounding the point of intersection of the number lines. Naming Points Figure 0. shows the Cartesian plane, which is also called the (x, y)-coordinate plane. Notice how drawing lines through each integer value on both the x- and y-axes forms a grid. Thus drawn, the intersections mark points in the plane. Each point defines a unique location in the plane with an x component and a y component, called the x- and y-coordinates of a point. The two coordinates are given together as the ordered pair (x, y). The point of intersection of the zeros has a special name, the origin, (0, 0). The idea of plotting mathematical relationships using x-y coordinates called ordered pairs was formulated around 630 by Pierre de Fermat and René Descartes, mathematicians who worked separately on the same concept. In any sketch of a Cartesian plane, the origin is labeled as (0, 0) and the horizontal and vertical axes with x and y, respectively. Plotting Points The x value in an ordered pair is positive in the right half of the plane and negative in the left half of the plane. Similarly, the second number in an ordered pair
2 7 CHAPTER 0 GRAPHING LINEAR EQUATIONS The Cartesian plane or twodimensional space is often called -space. is positive in the upper half of the plane and negative in the lower half. Thus, the ordered pair (3, 4) is the point in two-dimensional space that lies 4 units above the number 3 on the x-axis and 3 units to the right of the number 4 on the y-axis. It is at this location that we plot the point (3, 4). The Cartesian plane itself is divided into quadrants, which are naturally defined by the boundaries of the axes. The quadrants are commonly numbered I, II, III, and IV, starting in the upper right of the plane and proceeding counterclockwise. EXAMPLE 0.: FIGURE 0. The Cartesian Plane Plotting Points on the Cartesian Plane Plot the points (0, 0), (5, ), (0, ), ( 3, 4), ( 4, 0), (, 3), and (6, 4) on the Cartesian plane. Solution: Observe the location of the plotted points. FIGURE 0.
3 THE CARTESIAN PLANE 73 EXAMPLE 0.: For any point on the x-axis, the value of the y-coordinate is zero. This is because the y-coordinate indicates the number of units a point lays above or below the x-axis. If a point is on the x-axis, it is zero units above or below the x-axis. Likewise, for any point on the y-axis, the value of the x-coordinate is zero. This is because the x-coordinate indicates the number of units a point lies to the left or right of the y-axis. If a point is on the y-axis, it is zero units left or right of the y-axis. We will return to this idea when we consider equations of different types of lines. EXAMPLE 0.: Quadrant Identification State the quadrant or axis where each of the following points lie. Solution: Plotting Points on the Cartesian Plane (Continued) (0,0) (5,) (0,) (-3,4) (-4,0) (-,3) (6,-4) (0, 0) is the origin. (5, ) is in Quadrant I. (0, ) is on the y-axis. ( 3, 4) is in Quadrant II. ( 4, 0) is on the x-axis. (, 3) is in Quadrant III. (6, 4) is in Quadrant IV. EXERCISES 0. For the following points, identify the x- and y-coordinate a) ( 5, 5) b) (4, 0) c) ( 3, 7) d) ( 9, 3) e) (380, 74) 0. Give the ordered pair for each point marked on the coordinate plane: FIGURE 0.3
4 74 CHAPTER 0 GRAPHING LINEAR EQUATIONS 0.3 Plot the following points on the Cartesian plane. Make sure each point is labeled with the correct ordered pair. a) (0, 0) c) (0, 7) e) (3, ) g) ( 6, ) i) ( 7, 7) b) (, 5) d) (, 3) f) (8, 0) h) (4, 8) j) ( 5, 5) 0.4 State the quadrant or axis where the following points lie. a) (, ) f) (34, 7) k) (0,0) b) ( 8, 6) g) (5, 5) l) (, +) or ( x, +y) * c) ( 0, 35) h) ( 4, 0) m) (+, ) or (+x, y) * d) (3.476,.36) i) ( 5, 5) n) (, ) or ( x, y) * e) (0, 3) j) (0, ) o) (+, +) or (+x, +y) * * x and y are positive real numbers 0. Graphing Points of a Line Our discussion of the methods for graphing linear equations in two-dimensional space begins with the line y = x. This equation describes the set of points with identical x- and y-coordinates. Notice that the origin (0, 0) is such a point. So are (6, 6), ( 4, 4), (, ), and ( 7, 7). Clearly, there are infinitely many 3 3 such points in two-space. Use graph paper when learning to graph lines. Some of these points are plotted in Figure 0.4. FIGURE 0.4 A plot of points with equal coordinates The line containing these points and all other points with equal x- and y-coordinates is given by the equation y = x and shown in Figure 0.5.
5 GRAPHING POINTS OF A LINE 75 FIGURE 0.5 Graph of y = x Making a Table of (x, y)-values The traditional method for graphing lines and other figures is to generate a set of ordered pairs in the form of a table of x and y values. Begin by choosing an arbitrary value for x and substituting this value into the equation to produce the corresponding value for y. For a line, of course, only two points are required, but extra points are often chosen just as a check. Nonlinear equations require several carefully chosen points. EXAMPLE 0.3A: Using a Table to Make a Graph Graph the line y = x by generating a table of ordered pairs. Solution: x 0 y = x A good idea when graphing is to write three or four ordered pairs in a table of (x, y) values. FIGURE 0.6 From geometry we know that two points, and only two points, are needed to construct a line. By graphing more than two points we are more likely to detect a possible error in any of the pairs. An incorrect computation would result in a point lying off the line.
6 76 CHAPTER 0 GRAPHING LINEAR EQUATIONS EXAMPLE 0.3B: Using a Table to Make a Graph Make a table of (x, y) values for the line y = x 3. Graph the line. Solution: Taking care to watch the signs of the coefficients the following table is generated and the points plotted on the graph: x y = x FIGURE 0.7 EXERCISES 0.5 Determine if the following points describe a line (Hint: draw a graph). a) (0, 0), ( 5, 5), (7,8), (, ) c) (, ), (0, ), (-, 3), (, 0) b) (, ), (0, ), (, 3), (, 5) d) (0, ), (, 3), (, 3), (, 5) 0.6 Graph the following lines by first making a table and then using your points to draw the graph. a) y = 4x d) y = x 8 g) y = 5x + 3 b) y = x 3 e) y = --x + 5 h) y = x + 4 c) y = x + x f) y = -- i) y = 0x For the line formed by y = 0x 3 state whether the following points lie on the line. a) (0, 3) c) (, 7) e) b) (, 3) d) ( 4, 43) (, 8)
7 THE SLOPE OF A LINE The Slope of a Line The slope of a line is an indication of a line s steepness. It also indicates whether the line is rising or falling as it progresses from left to right. Between any two distinct points, the change in y, or vertical displacement of the line, is called its rise. The corresponding change in x, or horizontal displacement, is called its run. The slope of a line is the ratio of the rise to the run and by convention it is usually represented by the letter m. This ratio can be calculated between any two distinct points on a line. For a given line, the slope does not change, regardless of which two points are used. Consider the points labeled P (x, y ) and P (x, y ) on the line in Figure 0.8. m = y x P(x, x ) is a common way to label a point whose coordinates are not specified. FIGURE 0.8 Slope of a line The slope, m, is calculated as follows: rise change in y y y y m = = = =. run change in x x x x Reversing the order of the subscripts does not change the computation. That is, Changes in y and x are indicated by y and x, read delta y and delta x. The Greek letter delta,, signifies the amount by which a variable changes. m y y ( y y) y y = = =. x x ( x x ) x x However, corresponding subscripts must be lined up, or the sign of the slope will be in error.
8 78 CHAPTER 0 GRAPHING LINEAR EQUATIONS EXAMPLE 0.4: Finding Slope Given Two Points Find the slope, m, of a line containing the points ( 6, ) and (, 8). Solution: Let ( x and, y ) = (,8) ( x, y ) = ( 6, ) y y Then m = = = =. x x Slope: Positive, Negative, Zero, and Undefined A line with a positive slope rises as it progresses from left to right. A line with a negative slope falls from left to right. A horizontal line has zero slope; it neither rises nor falls. In terms of rise over run, a line with 0 slopes paid to have 0 rise and its y = 0. A vertical line has an undefined (or infinite) slope. In terms of rise over run, a line with infinite slope is said to have 0 run and y =0. Point-Slope and Slope-Intercept Forms of a Line The point-slope form of a line is derived from the slope formula. We begin by considering a line with a known point called P(x, y ) and any other arbitrary point called P (x, y). y y The slope of this line is m = as shown in Figure 0.9. x x FIGURE 0.9 Point-slope form of a line The point-slope form of a line is determined by multiplying both sides of the slope equation by the quantity ( x x ) and rearranging terms: y y m = x x
9 THE SLOPE OF A LINE 79 Point-slope form of a line: m ( y y ) ( x x ) = ( x x ) ( x x ) y y = m( x x ) The point-slope form of the line allows us to find the equation of a line given one point and the line s slope, as illustrated in Example 0.5. EXAMPLE 0.5: Point-Slope Form of a Line Write an equation for a line with slope 3 passing through point ( 5, 7). Solution: Using the equation for the point-slope form of the line, (y y )=m(x x ), with (x, y ) = ( 5, 7) and m = 3, we get y 7=(x ( 5)) y 7= 3( x ( 5)) Point-slope form of a line? Point-slope form: y 7= 3x +5 We can further simplify the point-slope equation by rearranging terms as follows: Slope-intercept form: y = 3x 5+ 7 y = 3x 8 If two points on a line are given, the equation of the line can be found using the slope-intercept form as demonstrated in Example 0.6. EXAMPLE 0.6: Slope-intercept Form of a Line Write an equation for a line passing through points (6, ) and ( 4, 4). Solution: This example illustrates a common applied problem in mathematics in which several pieces of data are known (in this case, the coordinates of two points), and an equation that models the data is needed. The solution involves finding the slope between the two points and using one of the data points to establish the equation of the line. STEP : Find slope, with ( x, y ) = (6, ) and ( x, y ) = ( 4,4). rise change in y y y y m = = = = 4 ( ) 6 3 = m = = = run change in x x x x STEP : Substitute one of the points and the value of m found in the first step into the point-slope form of the line. Using (6, ) gives 3 y ( ) = ( x 6) 5 3 y+ = ( x 6) 5 Point-slope form The equation y = 3x 8 is the slope-intercept form of the line. This form is very useful for graphing linear equations. 3 8 y = x y = x+ 5 5 Slope-intercept form
10 80 CHAPTER 0 GRAPHING LINEAR EQUATIONS In general, the slope-intercept form of a linear equation is written as: y = mx+ b where m is the slope of the line, b is the y-intercept, and variables x and y are the point coordinates that comprise the line. The y-intercept, b, is where the line crosses the y-axis. The y-intercept occurs where the x-value is zero. The slope-intercept form of a line is particularly useful for graphing purposes. One can look at an equation in this form and using just the values of m and b, graph the line. First, find the value of b. Since the constant b represents the point where the line intersects the y-axis, it can be determined by substituting zero into the equation for x. The resulting value is the y-intercept, b: y = mx+ b= m(0) + b= 0 + b= b Thus, when x =0, y = b, so one point on the graph of y=mx+b is (0, b). We can graph this point directly from the equation by plotting the point (0, b) on the y-axis as shown in Figure 0.0.! Given an equation for a line in the slope-intercept form, y = mx + b, begin graphing the line at the point (0, b) and use the slope, m, to find a second point. FIGURE 0.0 y-intercept of a line Now that the value of b is known, use the slope, m, to draw the graph. Since we can read this value for a particular line directly from the equation, we can use it as a kind of map to find a second point of the line. The two points are sufficient y to draw the graph of the line. In general terms, letting m =, the second point x is found by moving y units vertically from (0, b) and then moving x units horizontally as illustrated in Figure 0.. FIGURE 0. Using slope, m, to graph a line from y-intercept (0, b)
11 THE SLOPE OF A LINE 8 Vertical and Horizontal Lines As declared earlier the slope of a horizontal line is zero and that of a vertical line is undefined. How are these facts reflected in the slope-intercept form of a line? Consider horizontal lines first. Since the slope m of a horizontal line is zero, the slope-intercept form of a horizontal line would reduce to y = 0 x+ b= b. Thus, a horizontal line intersects the y-axis at (0, b), as would any line whose slope is defined. The equation of a horizontal line (y = b) indicates that the y- coordinate is equal to b for all values of x as demonstrated in Figure 0.. FIGURE 0. Horizontal line, y = b The equation, y = represents the line running parallel to the x-axis and intersecting the y-axis at (0, ). Other points on the line include ( 7, ), (8, ), and (9, ). Regardless of the value of x, the y value is. The slope is 0.The slope of a vertical line is undefined, owing to the fact that between any two points on a vertical line there is no change in the direction of x. The x-axis is a horizontal line whose equation is y = 0. y y That is, m = = =. x 0 As such, we cannot fit the equation of a vertical line into slope-intercept form. Analogous to horizontal lines (y = b), the equation of a vertical line is x = a. Here, a is the x-intercept of the line, as well as the value of the x-coordinate for all values of y as illustrated in Figure 0.3. Recall that a fraction whose denominator is zero is undefined. The symbol for this is the same as for infinity,. FIGURE 0.3 Vertical line, x = a
12 8 CHAPTER 0 The y-axis is a vertical line whose equation is x = 0. GRAPHING LINEAR EQUATIONS For example, x = 0 is a linear equation in which no y appears. This is the line intersecting the x-axis at (0, 0) and running parallel to the y-axis. Other points on this line include (0, 9), (0, ), and (0, 3). Regardless of the value of y, the x value is 0. EXERCISES 0.8 Find the slope of each line containing the indicated points. a) (3, 4) (7, ) c) ( 8, -6) (, 6) e) (x, y ) (x, y ) b) (, ) (5, 5) d) (40, 5) ( 3, 5) 0.9 State whether the indicated points from lines whse slopes are: increasing (left to right); decreasing (left to right); zero; or undefined (infinite slope). a) (0, 0) (6,) c) ( 7, 7) ( 4, ) e) (3, 5) ( 3, 5) b) (, 3) (0, 0) d) (0, 0) ( 7, ) 0.4 Applying Linear Equation Forms to Graphs We now look at how to recognize and use the different forms of linear equations to graph lines. Graphing a Line from Slope-intercept Form, Using Only m and b Graph the line of the equation y=x 4. What is the x-intercept of the line? (The point of intersection with the x-axis.) EXAMPLE 0.7: Solution: This equation is in slope-intercept form: y=mx+b where slope m = and y- intercept b = 4. To graph the equation from this information alone, begin at (0, 4), marking the point. A slope of indicates a ratio of rise over run equal to. From (0, 4), move unit up and unit to the right. Mark a second point. Connect the two points thus plotted to produce the graph of the line y=x 4. Label the line with the equation, as shown.
13 APPLYING LINEAR EQUATION FORMS TO GRAPHS 83 EXAMPLE 0.7: Graphing a Line from Slope-intercept Form, Using Only m and b (Continued) FIGURE 0.4 The x-intercept is the point at which y = 0. Substituting y = 0 gives 0 =x 4, x=4. So the x-intercept occurs where x = 4 at the point (0, 4) as can be seen on the graph. Putting a Linear Equation in Slope-intercept Form to Graph It Graph the line 3x +5y =0. EXAMPLE 0.8: Solution: Although we could make a table of (x, y) values to graph this line, a quicker method is to put the equation in slope-intercept form. 3x+ 5 y = 0 5 y = 3x y = x y = x+ 5 Now that the original equation is in slope-intercept form, we can see that the 3 y-intercept is b =, with slope m =. Begin by plotting the point (0, ) and 5 then move down 3 units and right 5 units and plot the point (5, ). The reason for moving down is that the change in y is negative. Connecting these two points gives us the graph of the line as shown in Figure 0.5
14 84 CHAPTER 0 GRAPHING LINEAR EQUATIONS EXAMPLE 0.8: Putting a Linear Equation in Slope-intercept Form to Graph It (Continued) EXAMPLE 0.9: FIGURE 0.5 Finding an Equation of a Line from its Graph Use the graph in Figure 0.6 to find the equation of the line in slope-intercept form. (7, 0) FIGURE 0.6 Solution: Choose two points and calculate the slope, m. Let ( x, y ) = (7,0) and ( x, y ) = (,4) then y y m = = = = x x 7 ( ) 8 Substitute either (7, 0) or (, 4) into y=mx+b and solve for b. Choosing 7 7 (7, 0) gives 0 = (7) + b= + b. Hence, b = or 3. Since m = and b =, the equation of the line in slope-intercept is y = x+ 3. 3
15 EXAMPLE 0.0: APPLYING LINEAR EQUATION FORMS TO GRAPHS Finding Two Points on a Line Given the Equation Name two points on the line y = x +7. Solution: One point is clear from the equation: It is (0, 7), since the y-intercept is 7. To find a second point, choose a value of x that is easy to substitute into the equation; say x =: y = x+ 7 () + 7 = + 7 = 5 y = 5 when x= Thus, a second point on the graph of the line y = x +7 is (, 5). Be sure to write the coordinates in correct order after finding them. 85 EXERCISES 0.0 For each equation, identify both the slope and the point where the graph intersects the y-axis. a) y = --x 5 d) y = 6 3x g) y = x + 4 j) y 3 9 b) y = 0x + -- e) y = + 8x h) y = 5 c) y = 4x + ( 7) f) y = x i) y = 36 x = x Given the point and slope, graph the following lines and write their equations in slope intercept form. a) (0, 0), m = 4 f) (, 0), m = 7 b) (0, ), m = 5 g) (0, 0), m = c) (0, 4), m = -- h) ( 3, ), m = d) (, ), m = i) ( 4, 3), m = e) (, 5), m = 0 j) (6, 0), m = 0. Write the following equations in slope intercept form, then find two points on the line. 3 a) x --y = 4 3 d) 3x 8y = 4 g) x + y = 3 j) b) x y = 9 3 e) 9x --y = 6 h) 4x y = 34 c) 5x + y = 0 f) x y = 4 i) 3x + 7y = x = -- 8
16 86 CHAPTER 0 GRAPHING LINEAR EQUATIONS 0.3 Use the graphs in Figures 0.7A 0.7F to find the equations of the lines in slope intercept form. b) a) FIGURE 0.7A FIGURE 0.7B c) d) FIGURE 0.7C FIGURE 0.7D
17 APPLYING LINEAR EQUATION FORMS TO GRAPHS 87 e) f) FIGURE 0.7E FIGURE 0.7F
18 88 CHAPTER 0 GRAPHING LINEAR EQUATIONS
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