Objectives:
1. to understand and explain the differnt parts of the slope-intercept form
2. to be able to find the x and y-intercepts of a line
3. to be able to write the equation of a line in slope-intercept form
Definitions:
y-intercept: the point where a line crosses the y-axis. It is written in the form (0, y).
x-intercept: the point where a line crosses the x-axis. It is written in the form (x, 0).
Slope-Intercept form: the equation of a line written as y = mx + b where m is the slope of the line and b is the y-intercept of the line.
A line can be written in slope-intercept form which is y = mx + b (memorize this formula) with m = slope which we learned how to calculate on day 2 of this chapter and b = y-intercept, the point where the line crosses the y-axis.
The y-intercept (b) can be found in several ways. The easiest way is to be given a point on the line in form (0, y). The points (0, 2), (0, -3), (0, 0), (0,.5) and (0, -9.123) would all be examples of y-intercepts.
The second way is to find it on the graph. The picture below has the graph of three lines. We will find the y-intercept,b, for each line.
First look at the purple line. Now, find where the purple line intersects the y-axis. It intersects the y-axis at 1. So, b would be the point (0, 1).
Did you get (0, 0) for the blue line and (0, -1) for the red line? If not, look back the picture and find you mistake.
The third way to find b is through substitution. There are times when you have a point on the line other than the y-intercept and the slope. However, we need b in order to write the equation of the line in slope-intercept form. So, how then do we find b when we have that information. Example 1 will explain what to do in this situation.
Example1: Given m = 3 and and that the point (2, 7) is on the line, find b.
Here, we take our information and substitute into y = mx + b and then solve for b. We know that m = 3, x = 2 ,and y = 7. Substitution gives us the equation: 7 = 3(2) + b. We know have a simple first degree equation to solve. First, multiply to get: 7 = 6 + b. Finally, subract 6 from both sides to get the answer, 1 = b. Now, write b as the point (0, 1).
We now want to write the equation of a line in slope-intercept form. In order to do this, we need values for m and b so that we can substitute them into the equation y = mx + b. So that when we are finished, the answer will look like y = 2x + 5. Note, that when writing the equation of the line that the x and y are not replaced by a number, only the m and the b are replced.
Example2: Write the equation of a line in slope-intercept form with m = 5 and b = -2. Here we already have all of the information that we need, so all we have to do is make the correct substitutions. So, we get the equation: y = 5x + -2 which we write as:
y = 5x - 2.
Example3: Write the equation of a line in slope-intercept form with a slope of .7 that goes through the point (0, 4). Here we already have m and have been given b but in the form of a point. Remember that b written as a point has an x-value of zero. Since we have the point (0, 4) that means that b = 4. Now, substitute the values of m and b to find the equation of the line to be:
y = .7x +4.
Example4: Write the equation of the line in slope-intercept form that passes through the point (2, 11) with m = 4. Here we are not given b, so we must find it. This will work just like example 1. Start by substituting for x, y, and m. This yield the equation: 11 = 4(2) + b. Then, multiply to get: 11 = 8 + b. Now, subtract 8 from both sides to find that 3 = b. Finally, write the equation of a line with m = 4 and b = 3 which is:
y = 4x + 3.
The other skill that we want to learn on this day is to find the x and y-intercepts of a line. In these cases, we will already have the equation of the line and try to find where the line intersects the x-axis and the y-axis.
So, we ask ourselves what we know about the x-intercept and the y-intercept. Well, we know that the x-intercept has the form (x, 0), or that y = 0 where the line intersects the x-axis. We also know that the y-intercept has the form (0, y) or that x = 0 where the line intersects the y-axis. This information will be used to find the intercepts.
To find the intercepts, we will do two different substitutions into our equation of a line. First, we will substitute x = 0 into the equation and solve for y. This will give us our y-intercept. Second, we will substitute y = 0 into the equation and solve for x. This will give us our x-intercept.
Example5: Find the x and y-intercepts of the line: y = 2x - 8 and write them as points.
Part 1: Finding the y-intercept.
So substitute x = 0 into the equation. This gives us y = 2(0) - 8. Do the order of operations to get, y = -8. This means that b = -8. So, it is the point (0, -8).
Part 2: Finding the x-intercept.
Here, substitu y = 0 into the equation. This gives us: 0 = 2x - 8. Add 8 to both sides to get 8 = 2x. Divide both sides by 2 to find that: 4 = x. This means that the x-intercept is at the point (4, 0).
Is that the only way to find the x and y-intercepts. Can you think of any easier way to find them? If so, bring in a written explanation of easier way to find one or the other or both for 5 bonus points.