I've been doing some summer practice assignments for my upcoming calculus class, and I have been tasked with graphing $y = (x^2 − 1)(x − 2)^2$ by hand.
At first I started making a table of values, but I quickly realized both $x = 1$ and $x =2$ have $y$ values of $0,$ with a relative maximum in between. I am struggling to figure out how to find relative min/max without the use of my calculator and without the use of differential calculus (thanks google) as I have not learned this yet. If you were in my shoes, how would you go about solving this? (Without the use of a calculator).
Best Answer
Finding the zeros is a good first step and that's really easy in your case: in addition to the ones you found there is also $x = -1$. You should also note that $x = 2$ is a double zero. Since the degree of the polynomial is 4, that's all the roots.
Finding the behavior at very large (positive and negative) values of x is the next step. In your case, the function goes to plus infinity at either end, because of the squares.
Finding the y-intercept might be helpful as well: that's the point where the graph crosses the y-axis. Just set $x=0$ and calculate the value of $y$ there.
Finding maxima and minima would be the next step but since you have not had calculus yet, you can't use derivatives. But you can see that your function is negative between $x=-1$ and $x=1$ because the first factor is negative there (the second factor is always positive - or zero). So you know that the function is positive when $x<-1$, negative when $ -1 < x < 1$ and positive when $x > 1$, except at $x=2$ where it has a double zero. At the double zero, the function just kisses the x-axis - that's true for a zero of multiplicity higher than 1 and the higher the multiplicity the flatter the curve is around that zero.
That should be enough to let you sketch the function.