Slope and Y-intercept

Shodor > Interactivate > Discussions > Slope and Y-intercept

Student: Why is there sometimes a number added on to the end of functions but at other times not?

Mentor: When a number is not added on to the end it is because the number equals 0.

Student: But what does that number do?

Mentor: It's called the Y-intercept which means that it is the point at which the function intersects the Y-axis.

Student: But I thought all points consisted of two coordinates: an X-value and a Y-value.

Mentor: You're right, but if we know a point is the Y-intercept what can we assume about X?

Student: Well...if it is on the Y-axis then X would be 0 wouldn't it?

Mentor: Exactly, this means that we only need one number because we already know that X is 0.

Student: But how do we know that it is the Y-intercept? Why can't it be the X-intercept?

Mentor: Well, the Y-intercept is the place on the graph where the function intersects with the Y-axis, so the X-intercept must be what?

Student: The place on the graph where the function intersects with the X-axis.

Mentor: Right, so knowing that we can now test for the X and Y-intercepts. Let's do Y first. To start you need to make up a linear equation.

Student: OK, Y=5X+3

Mentor: So, since we know that we are looking for the Y-intercept we can replace x with 0 because we already know that coordinate, right?

Student OK, so then it would be Y=5*0+3.

Mentor: Right, now can you simplify it?

Student: Well, then it would be just Y=3 because 5*0=0.

Mentor: Exactly! So, when X is 0, the function crosses the Y-axis at Y=3, which is exactly what was added on to our equation.

Student: OK, so when X=0, Y will always be the number added on to the end of the function. Will this work for non-linear functions too?

Mentor: Try it!

Student:OK, well...I'll use Y=X 3 +4X+6. So then I replace X with 0...and since 0 3 and 4*0 both equal 0 then Y=6. It works every time! But how do we know the number on the end isn't the X-intercept?

Mentor: Well, to find out you just need to replace Y with 0 this time. So make up a function and try it out.

Student: Okay, Y=4X+6.

Mentor: Good, now replace Y with 0 and solve the equation for X.

Student: Right, so 0=4X+6, so I would subtract 6 to get -6=4X, and then I would divide both sides by 4 so it is...-6/4=X.

Mentor: Good Job! You can now see that when Y=0, X does not equal the number added on. So now we know that "the number added on" is not the X-intercept, but what did we discover it was?

Student: The Y-intercept.

Mentor: Exactly, now that we know what that number is, what do you think would happen if we changed it?

Student: Well, the Y-intercept would change wouldn't it?

Mentor: Right, but what would happen to the entire line?

Student: Well, it would change because the equation has changed, right?

Mentor: Yes, and how would it change?

Student: I'm not sure. Maybe it would curve past the Y-intercept.

Mentor: Not quite right but you are thinking along the right lines. Lets go to Function Flyer and test it out. OK, enter X+1 in the "Set Function" bar next to f(x)=. What do you see come up?

Student: It is a diagonal line going through 1 on the Y-axis.

Mentor: Right, now do you see how the 1 in your equation is purple and how there is a purple slider bar down below it?

Student: Yes. Does this let me change the Y-intercept?

Mentor: Right, now play with the slider and see what happens to the line.

Student: The line moves up and down!

Mentor: What happens to the Y-intercept in your equation as you move the slider?

Student: It goes up and down. Whoa, it goes into the negatives. What does that mean?

Mentor: It just means that the Y-intercept goes below the X-axis on the graph into the negative numbers.

Student: Does this work for all types of functions?

Mentor: Yes, you can try any type of function you want and as long as it is a real function it will work.

Student: OK, I understand what the Y-intercept is and what happens when you move it, but what does the number next to the X in a linear equation mean?

Mentor: Good question. Whenever a number is next to a variable that means you multiply that number with the variable.

Student: OK, so 3X would be 3*X?

Mentor: Exactly.

Student: But what does that number mean in the equation? Is that number the X-intercept?

Mentor: No, it is not the X-intercept because as you'll remember from the problem we did earlier, we got -6/4 as the X-intercept, but the number next to the X in that problem was a 4. What that number actually is is the slope. Do you have any guesses as to what the slope is? Think of it like the slope of a hill.

Student: Well...then I guess it would be how steep the line is.

Mentor: Right, the slope tells us how many squares that the line goes up for every square the line goes over. The more common way to say that is rise over run. Rise meaning the amount the line goes up and run meaning the amount the line goes over in any section of the line.

Student: OK, so if you go up three squares every time you go over 2 squares then the slope would be 3/2?

Mentor: Perfect.

Student: But sometimes I see lines that go down from left to right instead of going up from left to right. What does that mean?

Mentor: Well, if you go down two for every one that you go over, do you have a guess as to what the slope would be?

Student: Um...well, the rise over run would still be 2/1 right?

Mentor: No, not quite because a line with the slope of two would be going up; the slope of a line going down would be just the opposite.

Student: Oh! It would be -2/1.

Mentor: Right, whenever you have a line going down from left to right than the line has a negative slope. Why don't we go to Function Flyer again and see what happens when you change the slope.

Student: OK, I'm at Function Flyer. What do I do now?

Mentor: Enter in a linear equation including a slope and a Y-intercept.

Student: OK. f(X)=6X+3. It said there was an error in my function!

Mentor: It is not actually your function that is wrong. In Function Flyer, you have to put a multiplication sign in between the slope and the variable.

Student: Oh, OK. So now it is f(X)=6*X+3. This time it gave me two sliders though.

Mentor: Right, the purple bar matches up with the purple number in your function and the green bar matches with the green number in your equation. In this case, purple is the slope and green is the Y-intercept. Try moving them around.

Student: As I move the slope up, the line gets steeper and as I move it down, it gets flatter.

Mentor: At what point do you think the line will be completely flat?

Student: Um...if the slope is 0?

Mentor: Right, and at what point will the line will be completely vertical?

Student: I don't know, let me try it on Function Flyer. It won't go completely vertical. Why is that?

Mentor: That is because if you remember about rise and run, even the highest number imaginable would still just be the amount that the line rises for every square it goes to the right.

Student: So how you graph a vertical line?

Mentor: To graph a vertical line you don't use Y= you use X=. Whatever number X equals is the place in the X-axis that the vertical line passes through.

Student: OK, so it is impossible to graph a vertical line in Y=mX+b form?

Mentor: Right. So, today we have learned what the Y-intercept and slope are and what happens when you change these numbers in the equation.

Student: I always thought that graphs were just simple lines. I never knew you could do so much with them.

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