Graphing Quadratic Polynomials
Date: 02/07/2001 at 17:58:35 From: Amy Gregory Subject: Quadratic Function I've been working on questions like y = x^2 - 8x + 15. I have to put the equation on some kind of graph, but I don't know how. Can you please help me? Thank you very much, Amy
Date: 02/08/2001 at 02:20:44 From: Doctor Ian Subject: Re: Quadratic Function Hi Amy, When you want to graph a function like y = x^2 - 8x + 15 there are two ways to do it: the hard way, and the easy way. The hard way is to start picking values for x, and computing corresponding values for y: x = 0, y = 0^2 - 8(0) + 15 = 15 x = 1, y = 1^2 - 8(1) + 15 = 8 x = 2, y = 2^2 - 8(2) + 15 = 3 and so on. The problem with this is that it can take forever, and sometimes the function wiggles in between the points you choose, so you end up drawing the wrong graph. The easy way is to factor the equation: y = x^2 - 8x + 15 = (x - a)(x - b) where a + b = 8, ab = 15 = (x - 3)(x - 5) What does this tell us? Well, for starters, it tells us that y must be zero whenever x = 3 or x = 5. (Do you see why?) So we have two interesting points right away: (3,0) and (5,0). A quadratic function like this is always shaped like a parabola. The big question is: Does it open toward the top of the graph, or toward the bottom? Well, we know that a parabola is symmetric. That means that the vertex (the lowest point of a parabola that opens up, or the highest point of a parabola that opens down) has to be halfway between x = 3 and x = 5 - that is, it has to be at x = 4. So we can evaluate the function at x = 4: y = 4^2 - 8(4) + 15 = 16 - 32 + 15 = 31 - 32 = -1 So the function has to look like this: | A = (3,0) | B = (5,0) |--|--|--|--A--|--B--|--|--| C = (4,-1) | | C We can complete the graph by sketching a parabola through these three points: | . . | | | . . | | . . |--|--|--|--A--|--B--|--|--| | . . | C Note that the 'easy' way requires you to _know_ more than the 'hard' way. A lot of what you're supposed to be learning about graphs is how to guess what you're going to see before you begin plotting. In particular, the degree of a polynomial function tells you how many times it will bend: A linear polynomial (y = ax + b) doesn't bend at all. A quadratic polynomial (y = ax^2 + bx + c) bends one time. A cubic polynomial (y = ax^3 + bx^2 + cx + d) bends twice. And so on. Once you know how many times it has to bend, then if you can find the points at which the function crosses the x-axis, it becomes easy to draw the function. That's what I did here. (Being able to do this kind of thing, by the way, is why they make such a big deal out of factoring.) I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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