Locating less obvious y-intercepts on graphs | Grade 8 (TX TEKS) | Khan Academy

So we have the graph of a line shown right over here, and my question to you is: what is the Y intercept of this line? Pause this video and see if you can figure it out yourself.

All right, now let's work through this together. So when we just eyeball it over here, we can see that the Y intercept, which is where the line intersects the Y axis, is right about there. It's between two and three, but we don't know what it is exactly, so we're going to have to do a little bit of algebra.

You might already remember that the equation of a line can be expressed as Y is equal to mx + b, where m is the slope and b is the Y intercept. So what we're going to do is use these two points they've given us to first figure out the slope of this line and then figure out the Y intercept, and then we will be done.

So let's figure out the slope. Slope, you might remember, is change in y over change in x. Now, we could do this two ways. It looks like you could eyeball this maybe, but I'll confirm this using actual numbers. If I go from this point, which looks like the point (4, 0), to this point right over here, it looks like our change in y is -2 when our change in x is positive 3. So change in y over change in x is -2/3.

Another way, just to make sure that I'm not eyeballing it incorrectly, is I could look at these two points which they've actually given us the coordinates for: change in y over change in x. From this point (-2, 4) to this point (7, -2), I could say my change in y is this y value (-2) minus this y value (4) over, and I didn't really need the parentheses there, over this x value (7) minus this x value (-2).

So this is going to be equal to, let's see: -2 - 4 is -6, and then 7 - -2 is the same thing as 7 + 2, which is 9. Which is, once again, -6/9, which is -2/3.

So we know that this equation has the form—let’s do this another color—Y is equal to (-2/3)x + b. Now, to solve for b, I could just substitute one of these points in here for x and y and just solve for b. I usually try to pick whichever one the math might be a little bit easier. They're close, but I'll go with this one.

So when x = -2, y is equal to 4. So let me substitute that in here. So y is equal to four, so 4. I just substitute this y with 4, is equal to (-2/3) times x, so times -2 plus b.

And so let’s see: this is 4 is equal to (-2/3) times -2, which is going to be positive 4/3 plus b. Now, if I want to solve for b, I could subtract 4/3 from both sides, and I will get b is equal to 4 minus 4/3.

All I did here is I subtracted 4/3 from both sides, and I swapped the two so I could put b on the left just because I like how that looks. And so this is going to be the same thing. Four, if I express it as thirds, is the same thing as 12/3 minus 4/3, which is equal to 8/3.

And 8/3 is the same thing as, let me scroll down a little bit, it's the same thing as, let's see, three goes into eight two times with the remainder two, so it's two and 2/3, which is exactly what it looks like we have right over here. Our Y intercept right over there is at 2 and 2/3. 2 and 2/3, and we're done.