Graphing lines using x and y intercepts
Graphing Using Intercepts
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- What we're going to do in this video is graph some lines
- using the x- and y-intercepts of the line.
- So first of all, a line is defined by two points.
- If you find two points-- if you're able to plot two points
- on a line, then you can connect them and keep going in
- both directions.
- And you will have drawn the line.
- What I'm going to do in this video-- the two points I'm
- going to pick are the intercepts of the lines.
- There's two of them.
- There's the x-intercept and then there's the y-intercept.
- These are the points where the lines
- intersect the x- and y-axes.
- And just to make it clear, this right here is the x-axis
- on my graph paper.
- That right there is my x-axis.
- It just keeps going.
- That's my x-axis.
- This right here, in the middle, going up
- and down, is my y-axis.
- That right there is my y-axis.
- I don't think you can see what I just wrote.
- That is the y-axis right there.
- So let me show you what I mean by x- and y-intercepts.
- So let's do part (a).
- I'll do it in a different color, not the same color as
- my intercepts.
- Part (a), we have the equation y is equal to 2x plus 3.
- So first of all, let's think about what happens when x is
- equal to 0.
- When x is equal to 0, what is y equal to?
- y is going to be equal to two times 0 plus 3.
- Well, 2 times 0, that's just 0.
- So y is equal to 0 plus 3 or y is equal to 3.
- So the point x is equal to 0, y is equal to 3, that
- satisfies this equation.
- Or it's on the line defined by this equation.
- So the point 0, 3.
- Let me plot that.
- The point 0, 3.
- Now notice, this point-- this is when x is equal to 0, this
- sits on our y-axis.
- We call this right here a y-intercept.
- It is when x is equal to 0.
- But it's called a y-intercept because it sits on the y-axis.
- This is where the line intercepts the y-axis.
- Now let's do the same for the x-axis.
- Let's set y equal to 0.
- So if y is equal to 0, we have 0 is equal to 2x plus 3.
- We could subtract 3 from both sides of that equation.
- And you get negative 3 is equal to 2x.
- We can divide both sides by 2.
- And you get negative 3/2 is equal to x.
- And negative 3/2, that's the same thing as negative 1 1/2.
- So we know that the point negative 3/2, 0, right?
- The second scenario, y is equal to 0.
- We know that this set of values of x and y also satisfy
- this equation.
- Or that this point also sits on the line.
- So negative 3/2-- remember that's the same thing as
- negative 1 1/2.
- That's right there. y is 0.
- So that's that second point we just figured out.
- When y is 0, we're dealing with our x-intercept.
- We're intercepting the x-axis.
- That is the x-intercept right there.
- If I wanted to draw the line, it'll look like this.
- Very roughly, I'll connect those points and then just
- keep going.
- Keep going in both directions.
- Keep going in both directions forever and then I will have
- drawn my line.
- Let's do one more of these.
- I don't want to make it too messy over here.
- Let me clear all this stuff out of the way.
- Let me clear this out of the way as well.
- You could always pause it, if you want to gaze at it-- oh I
- shouldn't have done that, I got rid of my-- well you can
- see still the axes.
- I'll redraw the axes.
- I'll redraw it right there.
- Good enough.
- That's one, my x-axis.
- That is my y-axis right over there.
- Let me do part (b). (b) looks like an
- interesting one right there.
- (b) has 6 times x minus 1 is equal to 2 times y plus 3.
- So let's look at the situation when x is equal to zero.
- When x is equal to 0, that equation up there becomes 6
- times negative 1.
- Right?
- Because this is 0.
- So 6 times negative 1 is negative 6 is equal to 2-- let
- me distribute this-- 2y plus 6.
- I multiplied 2 times the y and times the 3.
- And then we can subtract 6 from both sides, so that you
- get negative 12.
- Negative 12 is equal to 2y.
- I subtracted 6 from both sides to essentially move this 6 on
- to the left-hand side.
- You subtract 6 here, this disappears.
- You subtract 6 from here, you get negative 12.
- Divide both sides by 2, you get negative 6 is equal to y.
- So our first point that we know is on the line is a point
- x is equal to 0, y is equal to negative 6.
- That is our y-intercept.
- Now let's see what happens when y is equal to 0.
- Go back to the original equation.
- We have-- let me write that a little bit different-- when y
- is equal to 0, we have 6 times x minus 1.
- Let me just distribute that.
- So 6x minus 6 is equal to 2 times y.
- Well, y is just going to be 0.
- So it's going to be 2 times 3.
- This whole thing is going to be 3.
- It's going to be equal to 6.
- So let's subtract 6 from both sides-- or let's add 6 to both
- sides of this equation.
- I want to get rid of that right there.
- So I need to add 6 to make that 0.
- So we get 6x is equal to-- you add 6 to the right-hand side,
- you get 12.
- And you get x-- divide both sides by 6, you get
- x is equal to 2.
- So our other intercept is x is equal to 2, y is equal to 0.
- So we have x is equal to 2, y is equal to 0.
- Right there.
- That is our x-intercept.
- And then our actual line, we just connect the dots.
- It will look something like that.
- Obviously, you have trouble drawing a straight line, but I
- think you got the idea.
- Two points define a line.
- Let me do one more of these.
- I could probably do it up here, in this
- real estate up here.
- Let's do part d.
- You can do part c if you want extra practice.
- So part d, we have x plus y is equal to 8.
- This is actually very, very straightforward.
- When x is equal to 0, what is y equal to?
- If this is 0, all we have left is y is equal to 8.
- And then when y is 0, what is x?
- Well, you can just almost cover this up.
- Then you just get x plus 0 is equal to 8.
- x is equal to 8.
- So that was pretty straightforward right there.
- So we have one point, 0, 8.
- So 0, 8.
- That is the y-intercept.
- And then we have the point 8, 0.
- 8, 0.
- That is the x-intercept.
- And then we can connect the dots.
- It looks something like that.
- Now let's do this problem down here.
- At a local grocery store, strawberries cost $3.00 per
- pound and bananas cost $1.00 per pound.
- If I have $10 to spend between strawberries and bananas, draw
- a graph to show the combination of each I can buy
- and spend exactly $10.
- Let me draw my axes.
- Make sure that these are nice and filled in.
- So that remember is my horizontal access.
- But just for fun, instead of calling it the x-axis, I'm
- going to call it the strawberry-axis.
- Actually let me call it s, the s-axis, where s is for
- strawberries.
- Let s equal the number of strawberries.
- I think you know where I'm going.
- b will be number of bananas.
- Let me do my b-axis.
- I'm going to plot the number bananas on the vertical axis.
- So this right there, that is the b-axis.
- b for bananas.
- Strawberries cost $3.00.
- OK, this isn't the number of strawberries.
- This is the number of pounds.
- Let me clear that up.
- This is pounds of strawberries.
- We're not going by the number of strawberries.
- We're going by pounds of strawberries.
- This is pounds of bananas.
- All right.
- Strawberries cost $3.00 per pound, bananas
- cost $1.00 per pound.
- So how much am I going to spend?
- I'm going to spend $3.00 times the number of strawberries
- because they're $3.00.
- All right, I'm going to spend $3.00 times the number pounds
- of strawberries because it's $3.00 per pound plus $1.00
- times the number of pounds of bananas.
- And they say I have $10.00 to spend on both.
- So that's going to be equal to $10.
- So 3 times the pounds of strawberries plus the pounds
- of bananas are going to be equal to 10.
- 1b, that's the same thing as b, so I can rewrite this as 3s
- plus b is equal to 10.
- Now let's plot this.
- Let's look at the situation where I get no strawberries.
- So my pounds of strawberries are 0.
- What does this equation become?
- It becomes 3 times 0 plus b is equal to 10.
- That's just 0.
- So b will be equal to 10.
- So I would have the point 0 pounds of strawberries, I
- could get 10 pounds of bananas.
- So if I get 0 pounds of strawberries, I can get 10
- pounds, right there, of bananas.
- Now what about the other scenario?
- What about the scenario where I get no bananas?
- I get 0 pounds of bananas.
- Now let's substitute back here.
- We have 3 times my pounds of strawberries plus 0 pounds of
- bananas will equal 10.
- That's just a 0.
- Divide both sides by 3.
- I could-- if I get no bananas-- I can get 10/3
- pounds of strawberries.
- Or this is equal to 3 1/3.
- So it would have the point 3 1/3 pounds of strawberries, 0
- pounds of bananas.
- So 3 1/3 pounds of
- strawberries, 0 pounds of bananas.
- And then we can connect the line.
- This whole line I'm going to draw, I am just going to draw
- it in the first quadrant.
- Because I can't have negative bananas.
- I'm not going to sell pounds of bananas.
- And I can't have negative strawberries.
- We're not talking about selling strawberries.
- So I can only buy these things.
- This is the amount that I'm buying.
- So let me to connect the dots right there.
- This is neat because this line shows all of the possible
- combinations of pounds of strawberries and bananas.
- For example, this point right here-- I don't know if I'm
- drawing it exactly-- it looks like if I get about 5 pounds
- of bananas, I can get a little under 2 pounds of
- strawberries.
- That's what that tells me.
- If I get 3 pounds of strawberries, if I get exactly
- 3 pounds of strawberries, I can get 1 pound of bananas.
- Every point here represents a combination that
- I can get for $10.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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