In math, the rate of change is called the slope and is often described by the ratio rise

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1 Chapter 3 Equations of Lines Sec. Slope The idea of slope is used quite often in our lives, however outside of school, it goes by different names. People involved in home construction might talk about the pitch of a roof. If you were riding in your car, you might have seen a sign on the road indicating a grade of 6% up or down a hill. Both of those cases refer to what we call slope in mathematics. Kids use slope on a regular basis without realizing it. Let s look at an example, a student buys a cold drink for $0.50, if two cold drinks were purchased, the student would have to pay $.00. I could describe that mathematically by using ordered pairs; (, $0.50), (2,$.00), (3,$.50), and so on. The first element in the ordered pair represents the number of cold drinks, the second number represents the cost of those drinks. Easy enough, don t you think? Now if I asked the student, how much more was charged for each additional cold drink, hopefully the student would answer $0.50. So the difference in cost from one cold drink to adding another is $0.50. The cost would change by $0.50 for each additional cold drink. The change in price for each additional cold drink is $0.50. Another way to say that is the rate of change is $ we call the rate of change slope. In math, the rate of change is called the slope and is often described by the ratio rise run. The rise represents the change (difference) in the vertical values (the y s), the run represents the change in the horizontal values, (the x s). Mathematically, we write m = y 2! y x 2! x Let s look at any two of those ordered pairs from buying cold drinks, (,$0.50) and (3,$.50) and find the slope. Substituting in the formula, we have:.50! 0.50 m = 3! ".00 2 Simplifying, we find the slope is $0.50. The rate of change per drink is $0.50 EXAMPLE Find the slope of the line that connects the ordered pairs (3,5) and (7, 2)

2 To find the slope, I use m = y 2! y x 2! x Subtract the y values and place that result over the difference in the x values. 2! 5 7! 3 = 7 4 The slope is 7/4 Sec. 2 Point Slope Form of a Line Slope has been defined as the rate of change and we have mathematically written it as: m = rise run m = y 2! y x 2! x If we were to rewrite the formula leaving out the subscript 2, we would have the m = y! y x! x Now, if I played with that equation and multiplied both sides of the equation by denominator, x! x, I would have: m(x! x ) = y! y Using the symmetric property, we have y! y = m(x! x ) That is an equation of a line with slope m passing through the point (x, y ). It s called the Point- Slope form of a line. The reason for that is simple, you are given a point and a slope in the equation. What s nice about the Point-Slope form of a line is if you are asked to find an equation of a line, then all you need to know is a point and the slope, then plug that information into the formula Point-Slope Equation of a Line: y! y = m(x! x ). Now, anytime someone asks you to find an equation of a line, you can use that formula. Sometimes you are asked to find an equation of a line and the slope is not explicitly given to you. If that occurs, you must first find the slope, then plug the numbers into the equation. Remember, parallel lines have the same slope perpendicular lines have negative reciprocal slopes.

3 What s nice about the Point-Slope Form of an Equation of a Line is that it is derived from the slope formula. That means if you happen to forget it, all you need do is write the equation for slope, then multiply both sides of the equation by the denominator. While there are other equations of a line, generally, anytime I m asked to find an equation of a line, I would use the Point-Slope Equation. The other equations; such as Slope-Intercept and General Form of an Equation of a Line are best used to graph lines. The other equations for lines will follow directly from the Point-Slope Equation of a Line. To use the Pont Slope Form of an Equation, you must know at least one point and be able to find the slope. Let s look at a few. Find an equation of a line that passes through (, 4) and has slope 5. Using y y = m(x x ) y 4 = 5(x ) That is an equation of a line passing through (, 4) with slope 5. Find an equation of a line that passes through the points (2, 5) and (4, 3). First, we ll break out the formula for finding equations of lines. y - y = m (x - x ) We do have a point, so I know ( x, y ). The slope, m, was not given to me. Can I find the slope given the information? The answer is yes. The slope is the change in y over the change in x. Using the two ordered pairs (2, 5) and (4, 3) given in the problem, we have: m = 3! 5 4! 2 Now that I know a point and I just found the slope the slope, I just plug those into the Point Slope Equation. It doesn t matter which point you use. Using either point, you will arrive at the same answer. I ll use (2, 5). y - y = m (x - x ) y 5 = 4 (x 2) = 8 2 = 4

4 y 5 = 4x 8 y = 4x 3 Often times the slope is not just given to you, you need to find it as we just did. Things you need to remember are parallel lines have the same slope, perpendicular lines have negative reciprocal slopes. As soon as someone says find an equation of a line, no matter what comes next, you should write the Point Slope Form of a Line, and plug in the appropriate numbers. Try a couple. Find an equation of a line passing through (2, 3) and (3, 7). 2. Find an equation of a line passing through (4, 5) having a slope Find an equation of a line passing through (3, ) that is parallel to 4x 2y = 6. Sec. 3 Graphing Linear Equations In order to plot the graph of a linear equation, we solve the equation for y in terms of x, then we assign values for x and find the value of y that corresponds to that x. Each x and y, called an ordered pair (x, y), represents the coordinate of a point on the graph. EXAMPLE: 3x + y = 2 Solving for y, I subtract 3x from both sides. y = 2 3x When I assign values for x, I get these y values: y x y x

5 Rewriting as ordered pairs: (0, 2), (2, -4), (-, 5) as plotting the graph. When I connect those three points I get a straight line, called a LINEAR equation. I could have chosen any values for x and found the corresponding y. However, it is easier to choose a convenient number, like zero, one, or two. Choosing a number like 00 would make my graph a lot larger. Or, I could have chosen a fraction, but that can be messy. EXAMPLE: x 3y = 8 Again, I solve for y in terms of x x 8 = 3y x 8 = y 3 Assign values for x and find the corresponding y s. The ordered pairs (0,! 8 ), (8, 0), and (5, -) 3 represent the points on the graph. x y 0! 8 3 y x If we did enough of these problems we would see a quicker way of graphing linear equations. First, all linear equations are graphs of lines, therefore all we need do is graph two points. Second, we would see that the value of x when the graph crosses the y-axis is always zero. Look at the last two examples. What s the value of y when the graph crosses the x-axis? Look at the graphs we ve already plotted, when the graph crosses the x-axis, the value of y is zero. Let s look at another example, first we ll plot the graph as we did in the previous two examples, then we ll look at a short cut. EXAMPLE: 3x - 4y = 2 Solving for y, we get 3x - 2 = 4y 3x - 2 = y 4

6 Now, assign values of x and find the corresponding y s. Graphing each ordered pair, our graph looks like this: y x y x If we take the same problem and realize the graph crosses the y-axis when x = 0, watch what happens. Sec. 4 Slope Intercept form of a Line We were able to find Point-Slope Equations of a Line by manipulating the slope formula and rewriting it as:. y! y = m(x! x ) The Point-Slope Form of an Equation of a Line allows us to find equations of lines. If we manipulate the Point-Slope form, we will see other forms of equations of lines that result in graphing linear equations in two variables by inspection. If you were asked to find an equation of a line passing through (,8) having slope 5, we d plug those numbers into y! y = m(x! x ). y! 8 = 5(x! ) Solving for y, we have y = 5x + 3 If I graphed that, I would find the graph crosses the y-axis at (0, 3) and has slope 5. That means, I would locate the point the graph crosses the y-axis, called the y-intercept, and from there go up 5 spaces and over one. The graph always crosses the y-axis when x = 0. If I did enough problems like that, I would begin to notice the coefficient of the x is the slope and the constant is where the graph crosses the y-axis. That leads us to the Slope-Intercept Form of an Equation of a Line. y = mx + b m is the slope, b is the y- intercept

7 To use that Slope-Intercept to graph, use the y intercept as the first point on the graph and then use the slope to go up and over to find the second point. Find the y-intercept and the slope of y = 2x 5 The graph of that line would cross the y-axis at (0, 5) and has slope 2. To graph that, I would go to the y-intercept (0, 5) and from there, go up two spaces and over one. Find the y-intercept and slope of y = 3 5 x + 2 Without plugging in x s and finding y s I can graph this by inspection using the Slope Intercept form of a line. y = 3 x + 2 is in the y = mx + b form. 5 In our case, the y intercept is 2, written (0, 2) and the slope is 3/5. Graphing I have: Now, the slope is 3. So from the y intercept 2, go up 3 and over 5 and place a point. 5 Connect those 2 points, I have my line. The question comes down to; would you rather learn a shortcut for grouping or do it the long way by solving for y and plugging in values for x? Let s try one more.

8 Graph y = 2 x 3 The y intercept is -3, (0, 3). the slope is 2. First go to -3, then group and over 2 for the next point. That s the graph. On a separate sheet of paper, graph the following equations by finding the y-intercept and the slope.. y = 2x + 2. y = 3x y = ½ x y = 2x 4 5. y = 3x y = 4x y = 2/5 x y = 4x 9. y = 3x 2 0. y = 5 Sec. 5 General Form of an Equation of a Line Instead of solving for y, and having an equation like y = 2x 5, if we decided to put the x s and y s on the same side of the equality, that would result in another form of an equation of a line, called the General Form of an Equation of a Line.. Transforming y = 2x 5, we have y = 2x 5 5 = 2x y

9 2x y = 5 Ax + By = C If we studied enough graphs, we would see that the graph crosses the y-axis when x = 0. We would also see the graph crosses the x-axis when y is 0. That leads us to be able to find two points by just looking at an equation, which allows us to graph equations of lines written in the General Form by inspection. Ax + By = C, if x = 0, then By = C! y int = C B Ax + By = C, if y = 0, then Ax = C! x int = C A Upon further inspection, we would find the slope of a line in General Form can be found by - A/B. Just as importantly, it should be noted that the Point-Slope Form of a Line came directly from the definition of slope. The Slope-Intercept was derived from the Point-Slope, and the General Equation came from the Slope-Intercept. They are all linked. Try this example: 2x + 3y = 6 When x = 0, 3y = 6, therefore y = 2 When y = 0, 2x = 6 therefore x = 3 Plot (0, 2) and (3, 0) and you re done. That beats solving for y and plugging in values for x. EXAMPLE: 3x 4y = 2 When x = 0, the 3x falls out of the problem, so -4y = 2 or y = -3. When y = 0, the -4y falls out, so we get 3x = 2, or x = 4. Notice, (0. -3) and (4, 0) were ordered pairs doing the problem the other way. Look on the chart. All we need do is plot the point just like before and we re done. Graphing by inspection takes a lot less time than plotting points. However, if you don t take the time to learn the different terms of a linear equation, you will have no other choice but to solve for y and plug in x s that takes time & space. Last time we looked at equations like 3x 4y = 2 and found out the graph crosses the y-axis when x = 0 and crosses x-axis when y=0. Knowing that, st plugging in x = 0, we found y = 3. So we have the ordered pair (0, -3), then letting y = 0, we see that x = 4. So the x intercept is at (4, 0). Plotting those 2 points, we have:

10 If we connect them we get the line. Graph 2x + 5y =0 Again, finding the x and y intercepts by letting x and y be zero. We found the x intercept is 5, when y = 0, and the y intercept is 2 when x = 0. The a = ordered pairs are (5, 0) and (0, 2). Plotting & Connecting the points, we have: The form of the equation 2x + 5y = 0 is called the General Form of Equation of a line. It is written AX + BY = C. One more, you say. Graph 2x 3y = 6 Let y = 0; x = 3 Let x = 0; y = -2 Those are the x and y intercepts. Graphing, we have: Connect those 2 points and you re done.

11 Recognizing patterns is important in mathematics it means the difference between doing problems quickly without arithmetic mistakes of trudging through. It s also important to know vocabulary. We ve discussed where the graph of the line crosses the x-axis, we call that the x-intercept. Where the graph crosses the y-axis is called the y-intercept. The x-intercept occurs when y is zero. The y-intercept occurs when x is zero. Graph the following equations by finding the x and y intercepts. ) 2x + 3y = 6 2) 3x + 4y = 2 3) 5x 2y = 0 4) 6x 2y = 6 5) x + y = 6) x y = 7) 5x 2y = 7

Pre-Algebra Notes Unit 8: Graphs and Functions

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