Graphing by. Points. The. Plotting Points. Line by the Plotting Points Method. So let s try this (-2, -4) (0, 2) (2, 8) many points do I.

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Section 5.5 Graphing the Equation of a Line Graphing by Plotting Points Suppose I asked you to graph the equation y = x +, i.e. to draw a picture of the line that the equation represents.

. Connect the points with a smooth curvee or line. So let s try this with y = x +. The first thing I m going to do is pick several values for x.

Then I ll substitute those numbers into the equation to get the corresponding y values.

Now, to get the graph, wee have to plot those points and draw the smooth line connecting them. You cann see the results in the graph to the right.

Since there s only one line that passes through two points, that s enough to get you a correct result. In practical terms, it s a good idea to find a third point as a double check.

1 Section 5.5 Graphing the Equation of a Line Graphing by Plotting Points Suppose I asked you to graph the equation y = x +, i.e. to draw a picture of the line that the equation represents. plotting points method works like this: The Strategies Graphing a Line by the Plotting Points Method 1. Find the coordinates of several points that are on the line.. Graph the points from step (1).. Connect the points with a smooth curvee or line. So let s try this with y = x +. The first thing I m going to do is pick several values for x. It doesn t matter what values you pick but I think you ll find it easier if you pick values that aren t tooo close together. Then I ll substitute those numbers into the equation to get the corresponding y values. x values y values - y = - + = -6 + = -4 0 y = 0 + = 0 + = y = + = 6 + = 8 Coordinates (-, -4) (0, ) (, 8) That takes us through step (1) of the strategy. Now, to get the graph, wee have to plot those points and draw the smooth line connecting them. You cann see the results in the graph to the right. A reasonable question for you to ask at this point is, How many points do I need to find? For a linear equation, where we know the result is going to be a line, the smallest answer is two. Since there s only one line that passes through two points, that s enough to get you a correct result. In practical terms, it s a good idea to find a third point as a double check. If you can t draw a straight line through all three points then you made a mistake somewhere in your calculations. Keep in mind that that only works for linear equations. If you ve got a more complex equation like y = x then two points almost certainly won t be enough. Example 1 Graphing by Plotting Points Draw the graph of the equation y = x 6. First, we need to find several points that are on the line. I ll pick -1, 0 and 4 as my x values. x-values y-values y = -1 6 = -9 y = 0 6 = -6 y = 4 6 = 6 coordinates (-1, -9) (0, -6) (4, 6) Examplee Graphing by Plotting Points Draw the graph of thee equation y =. This onee is a little tricky since there isn t an x value in the equation. Don t lett that throw you. That equation is saying that, for every x value, the corresponding y value is. Putting that into a table like the one in Example 1 gives us: : x-values y-values coordinates - (-, ) 0 (0, ) 4 (4, ) 118

If we graph those points and draw the line connecting them we get this for our graph: If we graph

Graphing by Intercepts Think back to Section 5. for a minute.

going to give us the basiss for our second graphing method.

If you can find the x-- and y-intercepts of a graph then they represent two points on the line.

Strategies Graphing a Line by the Intercepts Method 1. Find the x- and y-intercepts of the line.

Example Graphing by Intercepts Draw the graph of the equation y x.

2 If we graph those points and draw the line connecting them we get this for our graph: If we graph those points and draw the line connecting them we get the graph below. Graphing by Intercepts Think back to Section 5. for a minute. In that section, we saw how to find the x- and y-intercepts of a graph. going to give us the basiss for our second graphing method. That s Graphing by intercepts is just a special case of the Plotting Points method. If you can find the x-- and y-intercepts of a graph then they represent two points on the line. If you graph them and draw the line connecting them, you ll have the graph of the line. Strategies Graphing a Line by the Intercepts Method 1. Find the x- and y-intercepts of the line.. Graph the two intercepts.. Connect those pointss with a smooth line. Example Graphing by Intercepts Draw the graph of the equation y x. Remember that to find the x-intercept we have to start by setting y = 0 then solvee for x. 0 x x x x That tells us that the point (, 0) is on the line. Now we can find the y-intercept by setting x = 0 and solving for y. y 0 So our second point will be (0, -). If we graph those points and draw the line connecting them we get the graph on the right. 119

Graphing by the Slope-Intercept Method If the equation is already in the slope-intercept form then you can get everything you need to draw the graph right from the equation.

That s the point where the line crosses the y-axis so you know that the line goes through the point (0, -). Instead of solving an equation to get the second point, we ll get it using the slope.

Going up and over gives us the right hand point in the graph to the right. Now that we havee two points, the graph of the equation is just the line connecting them.

$If the y-intercept is b then the line goes through (0, b). Graph that point.. Look at the line s slope as a fraction, e.g. /.$ Step one is the most important step in that procedure and the one where I see the most errors. Remember that, to use this method, you have to put the equation in the slope-intercept form first, i.e. the equation has to be solved for y.

Step one is the most important step in that procedure and the one where I see the most errors. Remember that, to use this method, you have to put the equation in the slope-intercept form first, i.e. the equation has to be solved for y.

3 Graphing by the Slope-Intercept Method If the equation is already in the slope-intercept form then you can get everything you need to draw the graph right from the equation. Say I gave you this linear equation y x The first thing you should notice is that the y-intercept is -. That s the point where the line crosses the y-axis so you know that the line goes through the point (0, -). Instead of solving an equation to get the second point, we ll get it using the slope. Remember that the slope is the rise divided by the run. In our case the slope is / so rise run This means that to get from our first point to a second one we have to rise units and run. Going up and over gives us the right hand point in the graph to the right. Now that we havee two points, the graph of the equation is just the line connecting them. Strategies Graphing a Line by the Slope-Intercept Method 1. Put the equation into the slope-intercept form if it isn t already.. Get the y-intercept from the equation. If the y-intercept is b then the line goes through (0, b). Graph that point.. Look at the line s slope as a fraction, e.g. /. Start at your point from stepp 1 and go up by the number in the numerator and overr by the number in the denominator. That s your second point. 4. Connect those points with a smooth line. Step one is the most important step in that procedure and the one where I see the most errors. Remember that, to use this method, you have to put the equation in the slope-intercept form first, i.e. the equation has to be solved for y. If you have it in any other form, you won t be able to get the slope and y-intercept just by looking at the numbers in the equation. There are a few pitfalls here that you shoud be aware of. Issues with the Slope-Interceptt 1. What if the slope is negative? Remember from our Method discussion of number lines that negative means go to the left or go down. So if I told you that the slope was -/ you wouldd start at the original point and go units down and units to the right. It doesn t matter which number you put the negative sign on. You couldd have interpreted it as / (-) and gone units up andd three units to the left and gotten the same graph. 10

$. What if the slope is an integer? For example, what would you do with y = + 1? The slope here is but we want it to look like a fraction. To deal with this just remember that = / 1.$ $This is an example where the slope is an integer so we should interpret it as the fraction / 1. That lets us see both the rise and the run part.$ A slope of / 1 means that the y value increases units for every 1 unit that the x value increases.

A slope of / 1 means that the y value increases units for every 1 unit that the x value increases.

and drawing the line connecting them. Example 5 Graphing Lines Graph the equation y + = 6. Thiss one has an extra step that we have to do first.

4 . What if the slope is an integer? For example, what would you do with y = + 1? The slope here is but we want it to look like a fraction. To deal with this just remember that = / 1. Thiss means you would go up units and over 1. Example 4 Graphing Lines Graph the equation y = x + 1. This equation is already in the slope-intercept form, so we know that its y-intercept is 1 and its slope is. Since the y-intercept is 1, we know that the line goes through the point (0, 1). This is an example where the slope is an integer so we should interpret it as the fraction / 1. That lets us see both the rise and the run part. A slope of / 1 means that the y value increases units for every 1 unit that the x value increases. If we take our point, (0, -1), and increase the y part by and the x part by 1 we get our second point: (0 + 1, 1 + ) = (1, 4) Now we can graph our line by graphing the two points, (0, 1) and (1, 4), and drawing the line connecting them. Example 5 Graphing Lines Graph the equation y + = 6. Thiss one has an extra step that we have to do first. Since the equation isn t in the slope-intercept form, we have to put it in that form, i.e. we have to solve it for y.. y 6 y 6 y 6 y 6 6 y y Now we have the equation in the form we need. The slope is -/ and the y-intercept is so the line passes through (0, ). The slope is -/ so we know that our next point is going to bee down and right from (0, ). That would be the point (0 +, ) or (, 0). I ve plotted both of those points on thee graph below and drawn the line connecting them. Exercises Graph the following lines using the Plotting Points method. 1. y = xx + 4. y = -5x + 7. x = 6 11

4. y = 1 5. y = 4 6. y + 4x = 18 Graph the following lines using the Intercepts method. 7. y = 4 8. y = / + 4 10. y + x = 0 11. y = x + 1 Graph the following lines using the Slope-Intercept method. 1. y = x 1 16.

Justify your answer. Section 5.6 Some Special Cases Horizontal and Vertical Lines Try finding the slope of the line through the points (1, 4) and (1, 7).

5 4. y = 1 5. y = 4 6. y + 4x = 18 Graph the following lines using the Intercepts method. 7. y = 4 8. y = / y + x = y = x + 1 Graph the following lines using the Slope-Intercept method. 1. y = x x + y = y = x + y = y = 4x 6 y + = 5 y x = 0 x + y = 4 Technical Writing 19. Explain the relationship between the Plotting Points method and thee Intercepts method. 0. How many points do you need to find to graph a linear equation using the Plotting Points method? Justify your answer. Section 5.6 Some Special Cases Horizontal and Vertical Lines Try finding the slope of the line through the points (1, 4) and (1, 7). Thee calculation goes like this: m But that s a problem. It s impossible to divide a number by 0. It s a special case that we refer to as being undefined. This is a unique situation that only occurss when you ve got two points like the ones in the figure to the right where they line up one over the other.. Notice that the line through the points is a vertical one. That s the key point you should keep in mind: The slope of all vertical lines is undefined. Now try this situation: Find the slope of the line through the points (4, ) and (7, ). That calculation would be 0 m Take a look at the graph on the right. This situation is sort of the opposite of the previous one. Instead of being on top of each other the points are side by side which will give us a vertical line. We can make a general statement here like the one we made for vertical lines: The slope of any horizontal line is 0. Finding the equation through one of these lines is probably the easiest case we ve seen so far. Take a look at the two points I used for the vertical line. Notice that their x-coordinates are the same. That s a give away that the line is vertical and it also gives you the line s equation: x = 1. That s all there is to it. The x value is always goingg to be 1 and the y part doesn t appear in the equation. Pick any number you like for the y part of the coordinate and the x value thatt goes along with it will be 1. We can do the same thing with our horizontal line. If you look at the points, you ll seee that the two y parts are the same. That immediately tells you that the line is horizontal and that its equationn is y =. 1

Section Graphs and Lines

Section Graphs and Lines Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity