PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Once we know the basics of graphing polynomial functions, we can easily find the equation of a polynomial function given its graph. To find polynomial equations from a graph, we first identify the x-intercepts so that we can determine the factors of the polynomial function. We can also identify the sign of the leading coefficient by observing the end behavior of the function.
Once you've got some experience graphing polynomial functions, you can actually find the equation for a polynomial function given the graph, and I want to try to do that now. So this one is a cubic. We're calling it f(x), and so, I want to write a formula for f(x). Now let me start by observing that the x intercepts are -3, 1, and 2. And remember, x intercepts give us the factors: -3 gives me x+3; one gives me x-1; and two gives me x-2. And you might also notice that the function is decreasing. Normally, a cubic finishes by going up to infinity. But this one's going down to negative infinity, so there's probably a negative coefficient in front.
So let's see how we did here. We can test it by seeing if it passes through -3 when we plug in 0. So f(0)= -3, -1, and -2. So this is 2x3, which is -6. OK, it didn't work. Well that is because remember, there are lots and lots of cubics that pass through these three x intercepts. Any multiple of this polynomial will pass through those three intercepts. So the best way to start is to start with the x intercepts and then a little parameter 'a'.
We'll figure out what 'a' is by using the y intercept, but just make it 'a' for now. It will work out. You will see that 'a' is going to be negative because the thing ends up going downward, but that will all work itself out. So let's plug in the y intercept: f(0)=6. So we get ax3. I'm sorry, not 6. That's what I got last time: -3; 3, -1, and -2. And again, I get 6a=-3. So 'a' is -1/2. So my function is f(x)= -1/2, x+3, x -1, and x-2.
And you can see that if we hadn't been given this point, or any other point other than the x intercept, we wouldn't be able to actually find this value of 'a', and we could make 'a' anything we want.
But when you're given another point, you actually can determine the exact value of 'a' and come up with the right formula in the end. Remember, start with the intercepts, put an 'a' here, and use another point to find the value of 'a'.