Graphing Lines with y = mx + b
Lesson 15
In lesson fifteen, in the warm-up, you will identify the slope and the Y intercept of the line described by each equation. In the last lesson, your students learned that the slope was the coefficient of X and that the Y intercept was the constant when the line was in the equation Y equals M X plus B.
Our solve problem, we are going to study the problem in S. Jennifer’s baby girl Elizabeth was born weighing four pounds. She was born five weeks early. She has gained one-half pound each week. The equation Y equals one-half X plus four represents her weight. After graphing the line, what would be her weight after four weeks? In studying the problem, we will first underline the question, what would be her weight after four weeks? We will also answer the question this problem is asking me to find. This problem is asking me to find the weight of Elizabeth after four weeks.
When we have the function Y equals two X plus one, we will identify the slope which is always the coefficient of X, when the equation is in slope intercept form of Y equals M X plus B. Our slope in this equation is two. If we want to write our slope as a fraction, we would write it as two over one. If we want to identify the Y intercept, it is the constant, or B in the form Y equals M X plus B. So our Y intercept is one. We know that our slope is the rise over the run, and it is equal to two over one. We know that our Y intercept is one, which means it will intersect or cross the Y axis at one, or zero comma one. So we will plot the point zero comma one, giving us a Y intercept of one and we will use our slope of rise two, run one, rise two, run one, to find other points on our line. Earlier we had discussed that if we had had the slope negative two over negative one, it would be the same as having the slope of two over one. You may want to discuss with your students that a slope of negative two over negative one, meaning down two and back to the left one will also give us a point on our line. Once we have identified our points, we can connect them with a straight edge and you have graphed your line. The line Y equals two X plus one.
In the equation Y equals negative X plus four, if we are going to identify the slope, we know that it is the coefficient of X. Our coefficient here is negative one because there is an understood one in front of the X. We know that we have one X. So our slope is actually equal to negative one, and we can write this as our rise over our run equals negative one over one. Negative one divided by one is negative one. If we want to identify the Y intercept, we know that it is this constant. We have Y equals negative X plus four, and if Y equals M X plus B, our B is four, so our Y intercept is four. We find the Y axis and we go to one, two, three, four and this is our Y intercept. The slope is negative one, or a rise of negative one and a run of one, which means we will go down one, but to the right one. Down one, to the right, down one, to the right, creating other points on our line. We can connect these points using a straight edge. A negative slope, the line falls from left to right. Earlier, we discussed our slope in terms of going in the opposite direction. We know that the slope is equal to negative one. If we had risen one, and gone back to the left one, this would also be a point on our line.
To graph the line Y equals one-half X minus three, we will first identify our slope. Our slope is the coefficient of X when the line is in the form Y equals M B plus B. Our slope for this line is one-half. It is already a fraction. We need to identify the Y intercept. In this case, we have a minus three, which is the same as plus a negative three, so our Y intercept is going to be negative three. We will plot a point at zero, negative three, because this is where the Y intercept is negative three. We will use our slope of one-half. Because our slope is equal to the rise over the run, we will rise one point, and run two points. Rise one, run two, rise one, run two in order to find other points on the line. We had discussed finding points in the other direction, and we said that one-half was equal to negative one over negative two also, because a negative divided by a negative is a positive. If we went down and to the left two units, down one unit and to the left two units, we see we are still on the line. This is the graph of the line for Y equals one-half X minus three.
If we are going to graph the line Y equals negative two-thirds X minus one, we will identify the slope as the coefficient of X when it is in the form Y equals M X plus B. Our slope is negative two-thirds. Our rise over the run is equal to negative two over three. When we identify the Y intercept, we know that it is the constant after, but the equation must be in Y equals M X plus B form, so because it is a minus one, we know that our Y intercept will be negative one. The Y intercept of negative one is a point on the Y axis at negative one, and to use our slope of negative two-thirds, we will go down two units and to the right three units, down two units and to the right three units. Using a straight edge, we can connect our points. Our line goes through the three points. We had discussed our slope of negative two-thirds also being the same as two over negative three. This would mean we rise two points, and go back one, two, three units, giving us another point on the line. This is the graph for Y equals negative two-thirds X minus one.
Here we have a horizontal line. We are going to name four points on the line. The first point is zero, three. Our second point on the line is going to be one, three. Our third point on the line is going to be two, three. And our fourth point on the line is going to be three, three. All of these points are similar because the Y coordinate is three in every single point. If we choose two of our points, one comma three and two comma three, we can calculate the slope between these two points. X one, Y one, X two, Y two. Our slope is three minus three over two minus one, and three minus three is zero. Zero divided by one is zero. Our Y intercept is three. The line crosses the Y axis at three. So if we were going to write the equation of the line Y equals M X plus B, if we plug in a slope of zero and three for our Y intercept, zero times anything is zero. So the equation of our line is Y equals three, because zero times X is zero and zero plus three is three. All of the Y coordinates are three. When you have a horizontal line, your slope will always be zero and the equation of the line will always be Y equals a number.
We are going to label four points on our line. We can label the point two, zero, two, one, two, two is a point on the line, and two, three is also a point on our line. Looking at the four points that we named, how are the points similar? In this case, we see that the X coordinates are all two. If we pick two points from the line, two comma one, and two comma two, we can label the points X one, Y one, X two, Y two. When we plug into our formula for finding the slope, we will have two minus one over two minus two. Two minus one is one, and two minus two is zero. You can never divide by zero. In this case, the slope of our line is undefined. The slope of all vertical lines is undefined. We do not have a Y intercept in our line, however, what do we see about our X coordinates? With no Y intercept, we are going to write the equation of this line as X equals two. The slope of all vertical lines is undefined, and all vertical lines are written in the form of X equals a number.
We will now go back to our solve problem. We have already S-ed the problem, and we know that this problem is asking me to find the weight of Elizabeth after four weeks. When we organize our facts in O, we will first identify the facts. Jennifer’s baby girl Elizabeth was born weighing four pounds, fact. She was born five weeks early, fact. She has gained one-half pound each week, fact. The equation Y equals one-half X plus four represents her weight, fact. We have to decide whether the facts are necessary or unnecessary. Jennifer’s baby girl was born weighing four pounds. This is a necessary fact, so we will write it down. She was born five weeks early. This fact is unnecessary, so we will cross it out. She gained one-half pound each week. Our fact, the equation Y equals one-half X plus four represents her weight, is also necessary. In L, we are going to line up a plan. We will not have an operation because we are going to graph the line. In order to graph the line, we will identify the slope, identify the Y intercept, graph the line, and then we will find the point where X equals four. In V, we are going to estimate our answer and verify your plan with action. Since we know that Elizabeth weighed four pounds when she was born, we know that our answer has to be greater than four. To carry out my plan, I will use the coordinate plane. We are going to identify the slope. The equation is Y equals one-half X plus four. Our slope is equal to one-half. Our Y intercept is four, because the equation is in Y equals M X plus B. We will graph our Y intercept at four, and we will use our slope to find other points. We will rise one and run two, rise one and run two. We needed to find the weight at four weeks, and we see that the point at four has a Y coordinate of six, so Elizabeth will weigh six pounds after four weeks. So our coordinate four comma six. In E, we have to ask ourselves, does your answer make sense? Our question was the weight of Elizabeth after four weeks. Our answer is six, so our answer does make sense. Is your answer reasonable? We check our answer against the estimate, which is greater than four. Six is greater than four, so our answer is reasonable. And is your answer accurate? You could have your students graph the line on a graphing calculator, or they could make sure that they have the correct slope and they have the correct Y intercept. We will write our answer in a complete sentence. Elizabeth will weigh six pounds after four weeks.
We will now answer the essential questions to close the lesson. Number one, can the equations for all lines be written in slope intercept form? The equations for all lines, diagonal and horizontal, can be written in slope intercept form. Equations for vertical lines cannot be written in slope intercept form, because there is no Y in the equation for a vertical line. What is the general equation for a horizontal line? Y equals a number. And question three, what is the general equation for a vertical line? X equals a number.