explain how you would graph without a calculator V(x) = –(x + 1)2(x – 3)3

Answer 1

Take all the coefficients and multiply together to arrive at -6. Put that aside for one second. Now, what is left is (x+1)(x-3). This is a second degree polynomial (or parabola) which also must touch the x-axis at both -1 and 3. This is because at these values, the a zero is created (check it out -1+1 = 0 and 3-3 = 0. When a zero is created, by the zero product rule, the function must be 0 meaning it touches x-axis. Also, since a parabola by nature is symmetrical, the center must be the mid point between -1 and 3 or x=1. If you plug 1 into the function, you get the vertex at (1, -4)So if the parabola consisted only of (x+1)(x-3), you now have a depiction.

Now..go back to the -6 factor. What does that do. The 6 makes every output 6 times bigger which has the affect of stretching it vertically by a factor of 6. That means the vertex is no longer at (1,-4) but (1,-24) The zeros remain because multiplying them by 6 does nothing. What about the minus sign. That takes every positive output and makes it negative and every negative output and makes it positive. So it flips the curve upside down around the x-axis. So there you have it. An upside down parabola with vertex at (1, 24) touching the x-axis at -1 and 3.