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Writing an Equation from a Graph

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Today we're going to talk about writing an equation from a graph. So we're going to start by reviewing the ideas of slope and intercepts, and then we'll do some examples, writing equations from graphs.

So let's start by reviewing the ideas of slope and intercepts of a line on a graph. And then we'll do an example, writing an equation in slope intercept form of a line.

So remember, the slope of a line just tells us how steep it is. And if we have any two points on that line, we can then use the slope formula to calculate the slope.

And we also know that the y-intercept of a line is where the line crosses the y-axis. And it's also where the value of x is zero. And we know that the x-intercept is where the graph crosses the x-axis, and it's also where the value of y is zero.

So let's see how we can write an equation in slope intercept form for this line. So I remember that in slope intercept form, y equals mx plus b. I need to know the slope of my line and the y-intercept. So I'm going to start by calculating the slope. And again to do that, I just need to know the x and y values of any two points on my line.

So I'm going to go ahead and use this first point, which would be negative 2, 1, and I'll call that my x1 and y1 values. And then I'm going to use this point, which is 1, 4, and I'll call that my x2 and y2 values. So now that I have two points on that line, I can go ahead and use my formula for slope, which is just the difference in my y values over the difference in my x values.

So plugging in my y values, y2 is 4, and y1 is 1. So this would become 4 minus 1. And then to find the difference of my x values, my x2 value is 1, and my x1 value is negative 2. So this is going to give me 1 minus negative 2. So simplifying this, 4 minus 1 will give me three, and 1 minus negative 2 is also going to give me three. So I found that the slope of my line is just 3 over 3, or one.

So now that I know the slope of my line, I can look at my graph to identify the y-intercept. And I can see that the point where my graph crosses the y-axis is at 3, so my y-intercept is going to be 3. So now I can substitute my slope and my y-intercept into my equation and that will give me y is equal to 1x plus 3, which I could also just write as y is equal to x plus 3.

So for this example I again want to write an equation in slope intercept form for this line. However, even though I have two points that I can use to calculate the slope, my line does not intersect the y-axis where I can see it. So I can't just look at my graph and determine the y-intercept. So I'm going to have to use my equation to figure out algebraically what the y-intercept is. So I'll show you how to do that.

So I'm going to start by using my two points to calculate the slope. So this first point is at 2, 5, and that will be my x1 and y1 values. And this point is at 4, 2, and that will be my x2 and y2 values. So now that I have my two points, I can go ahead and use my slope formula to calculate the slope.

So I know that my slope is going to be equal to the difference of my y values. So 2 minus 5 over the difference in my x values. So 4 minus 2. Simplifying this, 2 minus 5 will give me negative 3, and 4 minus 2 is going to give me 2. So I know that the slope of my line is negative 3 over 2. So I'm going to go ahead and start inputting that value into my equation written in slope intercept form, and then we'll figure out algebraically how to get the y-intercept.

So in my equation written in slope intercept form, I know that my slope is negative 3 over 2. And I don't know what my b, what my y-intercept is, but because I know a point on the graph, I can use those coordinates, the x and y values of that point, and substitute those in for the x and the y into my equation. And it doesn't matter which point we use, as lines at the point on the line. So I can either use the x and y values, 2 and 5, or 4 and 2. So I'm going to go ahead and use 2 for x, and 5 for y.

So substituting those values into my equation, I have 5 is equal to, again, my slope times my x value, which was 2, plus b, which is what I'm trying to find, my y-intercept. So simplifying this, negative 3/2 times 2 is just going to give me negative 3. And now to isolate my b variable, I'm going to go ahead and add 3 to both sides. This will cancel, and I'm left with 8 is equal to b. So my y-intercept of this line is 8.

So now that I know that the y-intercept is 8, I can use that and the slope that I had before to write my equation in slope intercept form. So that's going to give me y equals negative 3 over 2x, plus 8.

So for my last example I've got the graph of a line. And I want to find the equation for a line that is perpendicular to this line. So I'm going to start by calculating the slope of this line, and then I'll use that slope to calculate the slope of the line that's perpendicular to that, and write an equation for it in slope intercept form.

So the slope of this line I can calculate by using these two points. I know that the slope is going to be the difference of my y values, so 5 minus 2, over the difference of my x values, so 3 minus 2. And simplifying that, I see that my slope is 3 over 1, which is to say as just 3.

So if the slope of this line is three, then that means that the slope of the line perpendicular to that is going to be negative 1 over 3. And that's because, remember that slopes that are perpendicular, or lines are perpendicular to each other have slopes that are opposite and reciprocal. So instead of positive 3 over 1, we have negative 1 over 3.

So now that I have my slope, I can go ahead and write an equation in slope intercept form, using my slope of negative 1 over 3. And then I'll have my x. And my y-intercept could be any value, because we don't need a specific y-intercept. We just want to know an equation for a line that is perpendicular to this line. In which case, that line could be perpendicular, but go through any point on my y-axis. So I'm just going to go ahead and pick a y-intercept of 2. So the line of the equation y equals negative 1/3x plus 2 is going to be perpendicular to this line.

So let's go over our key points from today. The slope of a line can be found by locating any two points on a line, and using the coordinates in the slope formula. The y-intercept is the point where the line crosses the y-axis, at x equals 0. And the x-intercept is the point where the line crosses the x-axis, or at y equals 0.

And finally, lines that are parallel have slopes that are the same. And lines that are perpendicular have slopes that are opposite reciprocals of each other.

So I hope that these key points and examples helped you understand a little bit more about writing equations from a graph. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.