Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the slope value is away from zero. Another thing to keep in mind is that the slope is the same value from point to point (or between any two points on the line). This is called the average rate of change because it describes how much the line changes from point to point. There are three different ways to find the slope, labeled m. rise 1) m = this is the graphical way run y y1 ) Given two points ( 1, y 1 ) and (, y ): m = 1 3) Calculate how much y changes from point to point and compare that to how much changes from point to point. The third way requires a little eplanation. Let s say we re given the points (3, 6) and (5, 9) and we want to find what the slope of the line is between them. I m going to make a t chart. means change in and y means change in y, or how much and y change from point to point. y y increases by + 3 6 +3 y increases by 3 5 9 y 3 Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m = y y1 9 6 3 = =. I just prefer and y and will use it almost eclusively this quarter 1 5 3 because of its adaptation to polynomials in general. There are a couple of other things to notice about slopes. One is that if a slope is positive, it is going uphill from left to right; if a slope is negative, it is going downhill from left to right. In other words, in a positive slope, as gets bigger, y also gets bigger. In a negative slope, as gets bigger, y gets smaller. Since the slope above is positive, it is going uphill from left to right. Let s graph it! y 3 6 5 9 We can also use the slope to find more points on the same line. Keep in mind that the slope is y 3 + 3 = =, so that if we + increase y by 3 and increase by, we get another point!
rise Notice how, as we go from point to point on the graph, we can identify the. If we go from run (3, 6) to (5, 9), we get y = +3 and = +, so, again, y = though, see how we get the same thing? This time y = -3 and = -, so This works because of the nature of ratios! 3. If we go from (5, 9) to (3, 6), y 3 = = Practice - Find the slope of the line between the pairs of points. 1) (-, -7) and (1, -1) ) (3, ) and (-3, ) 3) (, 5) and (, 3) 3. y - -7 1-1 y 3-3 y 5 3 m = y = m = y = m = y = One thing to notice about # is that with a slope of 0, our line is going to be horizontal. See how it s flat and has no incline? In #3, we have an undefined slope, which gives us a vertical line. In this case, our line is so infinitely steep that we don t have a number to represent its incline. We can also eamine the slope of a line based on the given equation of the line. For eample, let f() = 3 +. To find points on this line, we re not going to use the standard -, - 1, 0, 1, like we did for graphing basic equations. If we did that, we d end up having fractions as point values, and that s difficult to graph. It would be easier for us to pick -values that will cancel with that in the denominator. So, let s pick multiples of, like 0,, 8, and so on. f() = y = 3 + Also, eamine the slope from point to point using and y. What is the slope of this line? y y Do you see that number represented in the equation 0 y = 3 +? 8 Another thing to notice is that when = 0, the y-value is. Do you see the number represented in the equation y = 3 +?
The form y = m + b (when we solve for y ) is called slope-intercept form because if a line is in this form, the coefficient of is the slope and b is the y-intercept (the y-value when = 0 this is where the line crosses the y-ais). Notice how when we plug in 0 for, the m portion of the equation disappears and we re left with y = b. So, m is our slope and b is our y-intercept. Practice Find the slope and the y-intercept point on the following lines just by eamination: 1) y = 6 1 ) y = + 3 3) y = + 10 ) y = 7 5 8 point point point point The neat thing is that once we get one point, we can use the slope to find other points on the rise line, either by using the on the graph or by using and y on the t-chart. This also run means that if we know the y-intercept and the slope, we can find the equation of the line easily by simply plugging into our y = m + b format. Be aware, however, that this method only works if we are given a y-intercept, or a point where the -value = 0. It does NOT work if the point we re given isn t the y-intercept! Practice - Find the equation of the line having the given slope and passing through the given y-intercept. Notice that the given points all have = 0. 1) m = 3, (0, ) ) m =, (0, -3) 9 7 3) m =, (0, 9) ) m = 9, (0, -1) y = y = y = y = Another linear form that s important is what is commonly called standard form. It looks like this: A + By = C. The way we turn standard form into slope-intercept form is simply by solving for y. Practice Write the following in slope-intercept form ( y = m + b form). 1) 3 + y = 8 ) 5 7y = 1 3) + 3y = 7
Practice Graphing Lines that are in Slope-Intercept Form The lines below are given in slope-intercept form. Find the slope, y-intercept (b), and then graph the line. 3 1) y = + 3 ) y = 3) y = + 3 5 m = m = m = b = b = b = y y y The advantage to having an equation written in slope-intercept form when graphing is it immediately gives us a point (the y-intercept) and a slope, with which we can count out another point either on our -y grid or by using and y on a t-chart. But what if our line is given to us in standard form (A + By = C)? There are a couple of options. One option is to solve for y and rewrite our equation in slope-intercept form. The other option is to find the - and y-intercepts. In other words, plug in = 0 and find y and then plug in y = 0 and find. Eample: Graph 3 y = 1. Way #1: 3 y = 1 y = 3 3 Way #: 3 y = 1 m = b = 0 y 0 0 y Practice Graphing Lines that are in Standard Form Find the - and y-intercepts and graph the line. Also, use and y to find the slope of each line. 1) 3 + y = 6 ) y = y y
3) 5 + y = 10 ) 6 + 3y = 1 y y 5) y = 6 6) 3 + y = 3 y y In general let s solve the standard form of a line (A + By = C) for y so that we can see what it looks like in slope-intercept form. A + By = C What is the slope of the line in this format? Notice the negative sign! This means that we can look at any line in standard form and, just by eamination, figure out the slope. Practice Find the slope of the lines below. 1) 3 + y = 6 ) y = 3) 5 + y = 10 ) 6 + 3y = 1 5) y = 6 6) 3 + y = 3
A This method of finding the slope also works in reverse: if we know what the slope of B our line is, we can split the numerator and denominator into its A and B parts and find the front part of the equation of our line. We re looking at that net. Finding Equations of Lines Without a Y-Intercept There are several methods we can use to find the equation of a line. If we are given a slope and a y-intercept (where = 0), then we can simply insert those values into the y = m + b format and be finished. We have used this method previously. But what do we do if the point we re given is not the y-intercept? Well, we re going to use what we learned above, and we re going to avoid fractions! Eamples: Find the beginning of the equation of a line given the following slopes. 1) m = 3 ) m = 3) m = 7 1 ) m = 3 To actually find the entire equation, we need a point to go along with our slope. A slope alone is not enough because there are an infinite number of lines all having the same slope. Eample: The four lines on the grid to the right all have the same slope:. The difference is the value of the actual points the line goes through. We can see that difference most obviously in the values of the y- intercepts. Line #1 (top): y = + y-intercept = (0, ) Line #: y = + 1 y-intercept = (0, 1) Line #3: y = y-intercept = (0, 0) Line # (bottom): y = y-intercept = (0, ) So, if we are given a point in addition to our slope, then we can pinpoint the equation of the line we re looking for. We use the slope to find the beginning of our equation and the point to find the end of it! The eamples below use the same slopes as the first eamples in this section. Eamples: Find the equation of the line through the given point and having the given slope. 1 1) m =, (-, ) ) m =, (5, 6) 3) m =, (-1, -) ) m = 3, (3, -5) 3 7
Parallel Lines: The advantage of this method of finding equations of lines is evident where parallel lines are concerned. Parallel lines in a plane are lines that do not cross. They also have the same slope, so the left-hand side of the standard equation will, when simplified, be the same for all parallel lines. Eamples: All five of the following lines are parallel. Find the slopes of each line to prove this. 1 3 y = 3 y = 1 6 y = 7. 3 9 6y = 3 + y = 19 Eample: Find the line parallel to5 y = 10 which passes through the point (-1, ). Perpendicular Lines: Perpendicular lines are lines that cross in a 90-degree angle. In most cases, this means that if one line is going downhill (or has a negative slope), the other will be going uphill (or have a positive slope). The slopes also are reciprocals of each other. If one 3 slope, the slope of a perpendicular line will have a slope of. The only time this doesn t 3 apply is when one of the lines is either horizontal or vertical. In this case, the perpendicular line would be vertical or horizontal, respectively. Once we get the new slope, the rest of the process is the same: Eamples: Find the slopes of the lines perpendicular to the given lines below: 1 3 y = 3 y = 1 6 y = 7. 3 9 6y = 3 + y = 19 Perpendicular slopes: Eample: Find the line perpendicular to5 y = 10 which passes through the point (-1, ).
Given two points: Sometimes, instead of being given a slope and a point, we are given two points and are asked to find the equation of the line that passes through those points. While having two points is preferable in graphing, knowing the slope is preferable in finding the equation of the line. So, in this case, we ll have to find the slope first, and then use that slope and one of the two given points to find the rest of the equation. Eample: Find the equation of the line that passes through the points (-, -) and (1, 5). 1) Find the slope: y ) Put the slope into linear form: 3) Use (-, -): OR 3) Use (1, 5): (same answer, yes?) As you can see, the first two parts of this process are the same. Also, it doesn t matter which point you pick to plug in to finish finding your equation. I usually pick the numbers that are smaller and/or positive. If the slope is given to us, we can bypass part 1.