Effects of Change in Slope and y-Intercept
Lesson 20
In lesson 20, the warm-up will have your students identify the slope of each line and describe whether the line will be parallel or perpendicular or neither. This is important because in lesson 20, we are going to see how changing the slope and changing the line intercepts affects lines. Jim and Ray have joined two different health clubs. Jim pays $50.00 to join and then has $15.00 a month taken out of his checking account. The equation y=15x+50 represents the amount Jim pays for his membership. Ray pays $30.00 to start the club and then pays a monthly fee of $10.00. The equation y=10x+30 represents the amount Ray pays for his membership. If the equations were to be graphed, how would they be similar and different? We will underline our question, which is, if the equations were to be graphed how would they be similar and different? This problem is asking me to find how the lines are similar and different. We are going to graph our original equation y=x+2. We have a y intercept of 2 and a slope of 1, which means we will go up 1 and over 1 each time to find another point. We can connect our pointsand show the equation of the line. We want to compare a line with the value of -6 for b and we want to compare our lines to a line with the value of -5. I’m going to use two different colors for the two equations. Line A, we are going to change our value of b from 2 to -6, this will be y=x+-6. This equation could also have been written as y=x-6. We will have a y intercept of -6. Our slope stays the same. So we will find point using a slope of 1. And now we will compare the two lines. How does our orange line, y=x+-6 compare to our green line y=x+2. What happens to the y intercept? It moves from 2 to
-6, so it moves down. What happens to the x intercept? Our x intercept in green is at -2 and our x intercept in orange moves over to the right to 6. So, our x intercept moves to the right. We will now compare our original lines, y=x+2 to the line y=x+-5. We have a y intercept of -5 and our slope stays the same. We see that all of the lines are parallel because they all have the same slope. When we compare our second line, our line B in purple to our original line, y=x+2, what has happened to our y intercept of 2. It has moved down to -5. And what has happened to our x intercept of -2? It has moved to the right to 5. We will use the second graph to compare our original lines which we have a y intercept of 3 and 4. Our original line is y=x+2, with a y intercept of 2 and a slope of 1. We are going to compare the line, y=x+3. We will now have a y intercept of 3 and our slope stays the same; however, when we go from a y intercept of 2 to a y intercept of 3, the y intercept moves up. We also see that our x incept was at -2 and now it is out -3. So, when our y intercept moves up, our x intercept moves to the left. If we graph the line, y=x+4, we will have a y intercept of 4 and we will use a slope of 1. Once again the pink line, your y intercept moves up and your x intercept moves to the left. All of the lines are parallel which means they will never intersect because they all have the same slope; however, by changing the y intercept we are moving the line up and down on the y axis and this also changes our x intercept.
We want to compare the line y=-x+2 to other lines with a different value of b. We will first graph our original equation, y=-x+2 in green. We have a y intercept of 2 and a slope of -1, which means we will go down 1 and to the right 1 to find another point. We are going to compare this to the line, y=-x+4. We are going to change our y intercept from 2 to 4. Our slope still stays -1. If they have the same slope, we know they are parallel. We can see that if when change the y intercept from 2 to 4, the line moves up. We can also see, when we change the y intercept from 2 to 4, our x intercept moves to the right. Our third line, y=-x+-2 or y=-x-2; we will have a y intercept of -2; however, we still have the same slope. We connect the points and we see that the lines are parallel; however, when we change our y intercept from 2 to -2, our y intercept moves down and our x intercept moves to the left. To compare lines C and D to the original line. We will first graph our original line of y=-x+2. We will compare this to the line y=-x+3; our y intercept moves up to 3, however the slope stays the same, so the two lines are parallel. And our y intercept goes from 2 to 3, our line moves up; however when we change our y intercept from 2 to 3 our x intercept moves to the right. We also want to compare this to the line, y=-x-5 or y=-x+-5. We have a y intercept of -5 and a slope of -1. We see the line is still parallel; however, when we change our y intercept from 2 to -5, our y intercept moves down and our x intercept moves to the left.
We are going to use the graphs to compare the line y=2x+3 to other lines with a different slope. We will first graph our line, y=2x+3. Our line has a y intercept of 3. We want to graph the line y=-x+3 because we are changing the slope from 2 to -1. We will still have a y intercept of 3 however; our slope will now be -1. Comparing the 2 lines, how does the orange line compare to the green? When we compare it to the original equation, we see that the orange line is less steep. It does not fall as quickly as the green line rises. We also see that our x intercept shifts to the right. We are now going to compare the line, y=- ½ x+ 3. We have changed our slope from 2 to – ½. We will have a y intercept of 3 again and our slope of – ½ tells us we will go down 1 and over 2. When we connect our points, we see that the purple line does not fall as quickly as the green line rises so the slope is less deep. We also see that our x intercept of the purple line is to the right of the x intercept of the green line. To compare our last two lines, we will graph our original equation, y=2x+3. We are going to change our slope in line C from 2 to 3. We will still have the same y intercept of 3; however our slope now be 3, which means we will rise 3 and go over 1. The blue line compared to the original line, actually is deeper because it rises more quickly. If we look at our x intercept, the x intercept of the blue line is also further to the right. If we are going to graph the line, y= ¼ x+3, we will start with our y intercept 3 of and a slope of ¼ means we will rise 1 and go to the right 4. You can see that this line is less deep compared to the original line. You can also see that the x intercept is to the left of the original line.
We have underlined the question in our solve problem, and we know that this problem is asking us to find, how the lines are similar and different. In “O,” organize the facts, we must identify the facts. Jim and Ray have joined two different health clubs. FACT. Jim pays $50.00 to join FACT… and then has $15.00 a month taken out of his checking account FACT. The equation y=15x+50 represents the amount Jim pays for his membership FACT. Ray pays $30.00 to start the club FACT…and pays a monthly fee of $10.00 FACT. The equation y=10x+30 FACT... represents the amount Ray pays for his membership FACT. Now we have to decide which facts are necessary which is unnecessary. Jim and Ray have joined two different health clubs. Some students will say this is necessary, and some will say unnecessary; however, most should agree it is not to write it down on our list. Jim pays $50.00 to join… this is necessary. And then has $15.00 a month taken out of his checking account…this is also necessary. The equation y=15x+50 represents the amount Jim pays for his membership…this fact is necessary. Ray pays $30.00 to start the club…necessary. And then pays a monthly fee of $10.00…necessary. The equation y=10x+30…necessary…represents the amount he pays. We have determined that these are our necessary facts and now we are going to choose our operations. In this case, we are comparing the lines so, there really aren’t any operations taking place. When we line up the plan, we are going to compare the lines graphed. When we write in words what your plan of action will be, we will graph both lines and compare them.
In V we will verify our plan with action. The first step in V is to estimate your answer. We estimate that they will both rise from left to right, we know this because both of the slopes are positive so this will give us some idea of what the line should look like. Next we have to graph the lines. We first have to determine what scale we have to use on our graph. We know that we are comparing the number of months that they have been members of the gym and we are comparing this to the amount of money they have paid. A reasonable scale if we start at 0 for the x axis is 1, 2, 3,4,5,6,7,8,9,10 months. We are going to use a different scale on the y-axis because this represents our amount of money. This is the number of months and this is the amount of money paid, because we started with an initial cost of 30 and 50 dollars we are going to have to use a larger scale. We can use a scale of 10 so we can say 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. the first line that we are going to graph is the line that represents Jim, I am going to use a different color so you can see it. Y=15x+50. We start with an initial fee of 50 dollars and we have a slope of 15 which means we will go up 15 dollars and over for one month. By month two he will have paid 30 dollars plus the initial fee so we will be at 80 dollars and we will rise up 15 more and over to 95 dollars for three months. WE will use purple to represent Ray his equation is y=10x+30. We will start at 30 and each month he pays 10 dollars more. We will now discuss how the lines are similar. Both lines are going up from left to right so both lines have a positive slope. We can also discuss the differences in the two lines. Jim’s line in orange is steeper or rises more quickly than Ray’s line in purple. Jim’s line also has a larger y intercept then Ray’s line in purple. When we examine our results in E we will first ask ourselves does your answer make sense. We were supposed to ask our selves how the lines were similar and different and this is what we did so our answer makes sense.
We ask ourselves is our answer reasonable. When we compared our answer to the estimate we knew that they would both rise from left to right so yes our answer is reasonable. And is your answer accurate, your students need to make sure that they have a y-intercept of 50 and a slope of 15 for Jim and a y-intercept of 10 and a slope of 30 for Ray. We will now write our answer in a complete sentence.
The lines are similar because they both have a positive slope. The lines are different because Jim’s line is steeper than Ray’s line and Jim’s line has a larger y-intercept than Ray’s line.
We will close the lesson by looking at the essential questions. What happens to the graph of a linear equation when the slope is changed? The steepness of the line may change along with the direction positive or negative, and the x-intercept may change. Number 2 what happens to the graph of a linear equation when the y-intercept is changed?
The graph of the line shifts up if the y-intercept gets larger and shifts down if the y-intercept gets smaller.
In lesson 20, the teacher notes say you can use a graphing calculator in order to graph lines in the student pages. To graph the line, y=x+2 and y=x+-6, we will first press the y= button at the top of the calculator. Looking at our screen, we see that we have several y=, this allows us to graph more than one line at a time. The first line that we wanted to graph was y=x+2, so we will say x+2 to represent our first line. We will arrow down to the second y= and enter our second line, x+-6. We have now entered two lines on our y= screen. And, if we press the graph button, we will see the graph of the two lines, our first line and the second line. If your students are not paying attention, they may not be sure which line is which; the first line was graphed first and the second line is graphed second. If you go back to y= and you arrow down to your second equation, you can change the style of the line so that they can be sure which line is represented. Use the arrow key to go over to the line next to the y=. The first line is our normal line. If you press enter once, you see a thicker line. This will allow your students to distinguish between the two lines. If you press graph at this point, the first line will be a thin line and the second line will be thicker, so your students will be able to tell the difference between the two lines. If we go back to y= and we go down to the second equation and arrow to the left, we can run through the different options that your calculator has. If we press enter once in order to get a thicker line. If we press enter again, we have the shaded top half of a square or a triangle, and this is greater than, which we won’t use in this lesson. If we press enter again, we have a triangle with the bottom half shaded, this is less than. If we press enter again, we have a circle with a line; some people call this a little mouth because it looks like a mouth is running up. If I press graph at this point, you will see it graph the first line and then you will see that little circle crawl up your screen slower so that you can see the second line being graphed. Go back to y=, the circle with the line is your little mouth. If you press enter again, you will have a circle without a line so you see the mouth crawl up the screen; the second graph you will see the mouth but it does not leave the line. Go back to y= and our second equation. Press enter one more time and we will have a dotted line. The dotted line is often hard to see the difference between the solid line and the dotted line, just because of the screen. We go back to y= and go to our second equation and press enter one more time and we are back to our regular line. The thicker line is often a nice option for lesson 20, so that your students can distinguish between the two lines that you are graphing.