Then finding the slope, we can just use the same method that we have done the other ones we get the slope 4 1

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1 169 Graphing Equations with Slope Okay, now that you know how to graph a line by getting some points, and you know how to find the slope between two points, you should be able to find the slope of a line once you have an equation: Example: Find the slope and graph the line x-4y= Well, if we find a couple of points: (,1) and (6,4), the graph must look like this: (6,4) Then finding the slope, we can just use the same method that we have done the other ones we get the slope 4 1 m = = 6 4. (,1) Let s try it again with another line: = 5 First, find a couple of points (0,-5) is a nice easy one (7,-) is a little bit trickier Now to find the slope: = = Here I will do the work for another four lines. See if you can see the pattern: Equation: Slope: y = -x 5 m = - y = 7x + 4 m = 7 y = x 1 m = x y = m = 7 Can you see that? When y is by itself, the slope is simply the number in front of x. No change at all. Summary: When y is by itself, the slope is right in front of the x and the intercept is really easy to find: = + slope y-intercept

2 170 Since this is such a common ways of writing lines, it has been given the name of Slope-Intercept Form. Slope-intercept Form: - Written in the form = + - is the slope without any adjustment. - (0, ) is the y-intercept. Advantages: Slope is most easily found. Y-intercept is most easily found. Graphing is simple. Let s look at how nice this can be: Graph = 5 Based on our shortcuts, we have that the slope is = and that a nice point to start from is the intercept (0, 5). There is the graph: 7 Though slope-intercept is truly awesome, it is not the only way that equations of lines can be written. Another common way of writing the equation of a line is called Standard Form. This version has no fractions and keeps all the x s and y s on the same side of the equation. Here is an example: 4 = (0,-5) Since both Slope-Intercept and Standard form will be given to you to graph, you should be able to work with both of them. Important Note: You should also see that we can change from Standard form into Slopeintercept form (and vice-versa) simply by solving for y. In the example x-4y=, we get: 4 = Subtract x from both sides

3 171 Which is now in Slope-Intercept Form. Notice how the same line can be written in the different forms: 4 = + 1 = 4 Standard: x-4y= Slope: m= 4 y-intercept: (0, - 1 ) x-intercept: (,0) Divide both sides by -4 1 Slope-intercept: y= 4x - Slope: m= 4 y-intercept: (0, - 1 ) x-intercept: (,0) Here are some more that have been transformed into Slope-Intercept with the slope as well listed off to the side Standard Slope-Intercept Slope: 5 = 10 = 5 m = + 9 = 4 = 9 +4 m = 9 I hope that you kind of see a pattern emerging that you would be able to use as a shortcut. 5 + = 15 = m = 5 = 1 = 1 4 m = For any Standard Form equation written like + = The slope can be found easily by: In any case, you will learn and have practice with both forms. Being able to pick out intercepts and slope from lines will help you to graph them quickly. Having the slope especially makes it a cinch to graph lines. You only need to find one point, then follow the slope to the next point and draw the line. Example:

4 17 Graph the line and find the slope of y=- 5 x - 4 Well the slope is right in front of x, so m= - 5 One easy point is to stick in zero for x. We get the point (0,-4). Following the slope, (it is negative, so we will head down as we go to the right) down 5 over and we come to the point (,-9), and then draw the line. 7 Another example: Graph the line and find the slope of 7 =4 Well the slope is the opposite of over -7, so = 7 = 7-5 It appears that the easiest point in this one is the x- intercept, so we stick in zero for y and get x=: (,0). Following the slope we move up and over 7 to the next point (9,), and then draw the line. What if we are missing x or y? That covers graphing and finding the slope for the vast majority of equations. As you will recall, there were a couple of special cases where either the x or the y were missing. We now look to find the slope of these. We will work two examples of this: First: y= - Remember how to find a couple of points that work: (,-) and (- 1,-). It gives us the graph of a horizontal line where y is always - : Putting those two points in to the formula for finding slope, we get: 0 m = = = which means that all horizontal lines will have a slope of 0.

5 17 Second: x=5 Remember how to find a couple of points that work: (5,) and (5,6). It gives us the graph of a vertical line where x is always 5: Now if we put the points in the slope formula, we get: 6 4 = = bad news. (Division by zero is undefined.) which means that all vertical lines have undefined slope. m = undefined m = 15 For good emphasis, we restate the overview of slopes from Section 4.. To get a feel for slope a little bit better, we are going to take a little time to look at some slopes. You will m = m =1 = = =0 notice that the higher the number, the steeper it is. Common sense from that will tell you that a slope of 0 will belong to a line that is completely flat. Also, you should see that since numbers get bigger as the slope gets steeper, the slope of a vertical line would have to be far greater than a billion. On the other hand, numbers get increasingly large in the negative direction for lines that are heading down ever steeper. That means that vertical lines would have to have a slope that is less than negative one billion. Hmmmmmm. greater than a billion and less than negative a billion at the same time. No wonder that division by zero can t be done and is undefined. m = undefined = 8 A word of caution: Since the term no = 1 slope is interpreted by some to mean = zero slope and by others to mean that the slope doesn t exist, we will simply avoid the term. A vertical line has undefined slope and a horizontal line has a slope of zero.

E. Slope-Intercept Form and Direct Variation (pp )

E. Slope-Intercept Form and Direct Variation (pp ) and Direct Variation (pp. 32 35) For any two points, there is one and only one line that contains both points. This fact can help you graph a linear equation. Many times, it will be convenient to use the