Polynomial and Rational Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to graph a transformation for one of our basic power functions. Let’s graph y equals -2 times the quantity x plus 3 to the 4th power. Now here the basic functions y equals x to the 4th, so I’m going to make my initial table for u and u to the 4th.
Let’s use as our key points -1, 0, and 1 for starters. These are really easy tot raise to the power of 4. -1 to the 4th is 1, zero to the 4th is zero and 1 to the 4th is 1.
For our function x, -2x plus 3 to the 4th, we make the substitution u equals x plus 3 and that means that x equals u minus 3 and so in order to get our x coordinates we just take these guys and subtract 3, and that means that the whole graph is shifting to the left, so we have our x values. And now to get our y values all we have to do, we’ve already raise to the 4th power we just need to multiply the result by -2. So we take these values and multiply by -2. That gives us three nice points to plot for our graph of y equals -2 times the quantity x plus 3 to the 4th.
We start with (-4, -2), so -4 would be about here, -2, about there. (-3, 0), (-2, -2). Now let’s say you want to get a little more detail. You always have the freedom to plot more points if you want to so let me squeeze in -1/2 and positive ½. -½ to the 4th power is 1/16, ½ to the 4th power is also 1/16. So I can also translate these guys 3 units to the left and I get -3.5 for this guy and for this one again subtracting three I get -2.5 and for this, the y values, I multiply by -2 and I get negative 1/8, -1/8 . And so I can plot, negative 1/8 at both -3.5 and -2.5. -1/8, this is 1/2, that’s a ¼, -1/8 is really close to the axis about there and there. And that gives me enough points to draw a pretty good graph so let’s finish our graph.
There we go, a nice graph of y equals -2 times the quantity x plus 3 to the 4th power.