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