Video Transcripts: Intercepts Transcript

Intro to intercepts 

Let's say that we have the linear equation, y = 1/2x - 3. So if we wanted to draw the line that  represents the set of all points, all the coordinates where the x value and the y value satisfy this  equation, we could start off by just trying to draw, by trying to draw a few of those points, and  then connecting them with a line. Let's set up a little table here x, y. And we can just try a  couple of x values here, then figure out what the corresponding y values are. I'm going to pick x  values where it's going to be fairly easy to calculate the y values. Let's say when x is equal to  zero, then you're gonna have 1/2 x 0 - 3, well then y is going to be -3. When x is, let me try x = 2,  because then 1/2 x 2 is just gonna be 1. So when x = 2, you're going to have 1/2 x 2 = 1, -3 is -2.  When x is equal to, let's try 4. So 1/2 x 4 is 2, and then -3 is -1, and we could keep going but  actually all we need is two points for a line. So we're ready to plot this line if we'd like. The point 0, -3 is on this line. 0, -3 and actually let me do this in a slightly darker color so that we can see it on this white background. 0, -3 is on the line, 2, -2 is on the line. So 2, -2 and then we have 4, -1.  So when x is 4, y is -1, and I could draw a line that connects all of these so it would look  something like... If I, let's see if I could do this. It would look something like, it would something  like, like that. So this right over here, this is literally, this is the graph of y = 1/2x - 3. Now when  we look at a graph like this an interesting thing that we might want to ask ourself is where does  the graph intersect our axes? So first we can say, well where does in intersect our x-axis? When  you look at this, it looks like it happens at this point right over here. This point where a graph  intersects an axes this is called an intercept. This one in particular is called the x-intercept. Why  is it called the x-intercept? Because that's where the graph is intersecting the x-axis and the x intercept, it looks like this is at the .6, 0. Now it's very interesting, the x-intercept happens  when y = 0. Remember, you're on the x-axis when you haven't moved up or down from that axis which means y = 0. So your x-intercept happens at x = 6, y = 0. It's this coordinate. Now what  about the y-intercept? Well the y-intercept is this point right over here. This is where you  intersect or I guess you could say intercept the y-axis. So this right over here, that over there is  the y-intercept. The y-intercept is at the coordinate that has a 0 for the x-coordinate. X is 0 here and y is -3. X is 0 and y is -3. This was actually one of the points, or one of the pairs that we first  tried out. You can validate that 6, 0 satisfies this equation right over here. If x is 6, 1/2 x 6 is 3, -3  is indeed equal to 0. So now that we know what an x-intercept is, it's the point where a graph  intersects the x-axis or intercepts the x-axis and the y-intercept is the point where a graph  intercepts the y-axis or intersects the y-axis. Let's try to see if we can find the x and y-intercepts for a few other linear equations. So let's say that I had the linear equation. Let's say that I have  5x + 6y = 30. I encourage you to pause this video, and figure out what are the x and y-intercepts  for the graph that represents the solutions, all the xy pairs that satisfy this equation. Well the  easiest thing to do here, let's see what the y value is when x = 0 and what x value is when y = 0.  When x = 0 this becomes 6y = 30. So 6 times what is 30? Well y would be equal to 5 here. So  when x is 0, y is 5. What about when y is 0? Well when y is 0, that's going to be 0, and you have  5x = 30. Well then x would be equal to 6. Then x would be equal to 6. So we could plot those  points, 0, 5. When x is 0, y is 5. When x is 6, y is 0. So those are both points on this graph and  then the actual graph is going to, or the actual line that represents the x and y pairs that satisfy  this equation is going to look like, it's going to look like this. I'll just try. So I can make it go, it's  going to look like... It's going to go through those two points. So it going to...I can make it go 

the other way too. Let me see. It's going to go through those two points and so it's going to  look something like that. Now what are its' x and y-intercepts? Well, we already kind of figured  it out but the intercepts themselves, these are the points on the graph where they intersect the axes. So this right over here, this is the y-intercept. That point is the y-intercept and it happens,  it's always going to happen when x = 0, and when x = 0 we know that y = 5. It's that point, the  point 0, 5. And what is the y inter...what is the x-intercept? The x-intercept is the point, it's  actually the same x-intercept for this equation right over here. It's the point 6, 0. That point  right over there. 

X-intercept of a line 

The graph of the line 2y plus 3x equals 7 is given right over here. Determine its x-intercept. The  x-intercept is the x value when y is equal to 0, or it's the x value where our graph actually  intersects the x-axis. Notice right over here our y value is exactly 0. We're sitting on the x-axis.  So let's think about what this x value must be. Well, just trying to eyeball a little bit, it's a little  over 2. It's between 2 and 3. It looks like it's less than 2 and 1/2. But we don't know the exact  value. So let's go turn to the equation to figure out the exact value. We essentially have to  figure out what x value, when y is equal to 0, will have this equation be true. So we could just  say 2 times 0 plus 3x is equal to 7. Well, 2 times 0 is just going to be 0, so we have 3x is equal to  7. Then we can divide both sides by 3 to solve for x, and we get x is equal to 7/3. Does that look  like 7/3? Well, we just have to remind ourselves that 7/3 is the same thing as 6/3 plus 1/3. And 6/3  is 2. So this is the same thing as 2 and 1/3. Another way you could think about it is 3 goes into 7  two times, and then you have a remainder of 1. So you've still got to divide that 1 by 3. It's 2 full  times and then a 1/3, so this looks like 2 and 1/3. And so that's its x-intercept, 7/3. If I was doing  this on the exercise on Khan Academy, it's always a little easier to type in the improper fraction, so I would put in 7/3. 

Intercepts from an equation 

We have the equation negative 5x plus 4y is equal to 20, and we're told to find the intercepts of  this equation. So we have to find the intercepts and then use the intercepts to graph this line on the coordinate plane. So then graph the line. So whenever someone talks about intercepts,  they're talking about where you're intersecting the x and the y-axes. So let me label my axes  here, so this is the x-axis and that is the y-axis there. And when I intersect the x-axis, what's  going on? What is my y value when I'm at the x-axis? Well, my y value is 0, I'm not above or  below the x-axis. Let me write this down. The x-intercept is when y is equal to 0, right? And  then by that same argument, what's the y-intercept? Well, if I'm somewhere along the y-axis,  what's my x value? Well, I'm not to the right or the left, so my x value has to be 0, so the y intercept occurs when x is equal to 0. So to figure out the intercepts, let's set y equal to 0 in this  equation and solve for x, and then let's set x is equal to 0 and then solve for y. So when y is  equal to 0, what does this equation become? I'll do it in orange. You get negative 5x plus 4y.  Well we're saying y is 0, so 4 times 0 is equal to 20. 4 times 0 is just 0, so we can just not write  that. So let me just rewrite it. So we have negative 5x is equal to 20. We can divide both sides of  this equation by negative 5. The negative 5 cancel out, that was the whole point behind  dividing by negative 5, and we get x is equal to 20 divided by negative 5 is negative 4. So when  y is equal to 0, we saw that right there, x is equal to negative 4. Or if we wanted to plot that 

point, we always put the x coordinate first, so that would be the point negative 4 comma 0. So  let me graph that. So if we go 1, 2, 3, 4. That's a negative 4. And then the y value is just 0, so  that point is right over there. That is the x-intercept, y is 0, x is negative 4. Notice we're  intersecting the x-axis. Now let's do the exact same thing for the y-intercept. Let's set x equal  to 0, so if we set x is equal to 0, we have negative 5 times 0 plus 4y is equal to 20. Well, anything  times 0 is 0, so we can just put that out of the way. And remember, this was setting x is equal to  0, we're doing the y-intercept now. So this just simplifies to 4y is equal to 20. We can divide  both sides of this equation by 4 to get rid of this 4 right there, and you get y is equal to 20 over  4, which is 5. So when x is equal to 0, y is equal to 5. So the point 0, 5 is on the graph for this  line. So 0, 5. x is 0 and y is 1, 2, 3, 4, 5, right over there. And notice, when x is 0, we're right on  the y-axis, this is our y-intercept right over there. And if we graph the line, all you need is two  points to graph any line, so we just have to connect the dots and that is our line. So let me  connect the dots, trying my best to draw as straight of a line is I can-- well, I can do a better job  than that-- to draw as straight of a line as I can. And that's the graph of this equation using the  x-intercept and the y-intercept. 

Intercepts from a table 

The following table of values represents points x comma y on the graph of a linear function.  Determine the y-intercept of this graph. So just as a reminder of what the y-intercept even is, if  you imagine a linear function or a line if we're graphing it, if we imagine a line, so let's say that  is our line right over there. This is our y-axis. This is our x-axis. The y-intercept is where we  intersect the y-axis. Now, what do we know about the y-intercept? Well, at the y-intercept x is  going to be equal to 0. So this is the point 0 comma something. And so when people are talking about, what is your y-intercept? They're usually saying, well, what is the y-coordinate when x  equals 0. So we're really trying to figure out, what is the y-coordinate when x equals 0? So we  know the x-coordinate when y is equal to 0. So this is actually the x-intercept. So this point  right over here is the point 2 comma 0. So when people say x-intercept, that's the x-coordinate  when y equals 0. Well, they gave us the x-intercept. So that right over there is the x-intercept.  But what's the y-intercept? What is the y-value when x equals 0? Well, let's see. They give us  what happens to y when x is negative 2, when it's 1, when it's 2, when it's 4. So maybe we can  backtrack from one of these to get back to what happens when x is equal to 0. So let me  rewrite this table so I can give ourselves a little bit more breathing room. So let's say we have x  and we have y. x and y. And they already tell us that when x is negative 2, y is 8. And I actually  want to think about what happens when x is negative 1, when x is 0. Then they tell us when x is  1, y is 2. When x is 2, y is 0. This right over here is the x-intercept. When x is 4, y is negative 4. So  they skip 2 right over here. y is negative 4. So let's just see how y changes with respect to  changes in x. So when we go here, when x changes by 1, y goes down by 2. And it's a line, so it's  going to have a constant rate of change of y with respect to x. So similarly, when x increases by  1, y is going to decrease by 2. So y is going to be 6 here. When x increases by 1 again, y is going  to decrease by 2. So we're going to get to 4. And we see it works. Because if we increase by 1  again, then it is indeed the case that y decreased by 2. And you see here when we increase x by  2, then y decreases at twice the rate. Because now we didn't just increase by 1, we increased by  2. So now y is going to decrease by 4. And what's constant here is your change in y over your 

change in x. When x increases by 1, y decreases by 2. When x increases by 2, y decreases by 4.  Either way you think about it, your change in y for a unit change in x is going to be equal to  negative 2. But anyway, we actually answered the question before without even realizing it  when we filled in all of these values. What is the y-value when x equals 0? Well, the y-value is 4.  So the y-intercept here is 4. We didn't really graph this to scale. It would actually look a little bit  more like this if we were to try to graph it properly. So this right over here is 4. This right over  here is 2. And our line looks something like this. Our line will look something like that.