Graphing polynomial functions

Question

I am having problems understanding how to graph $(x^2 -1)(x^2 -9)$,

Any help would be much appreciated! Thanks

Answer 1

The first thing you should do is factor the polynomial completely. Here, we have $(x^{2} - 1)$ can be factored into $(x - 1)(x + 1)$ because it is a difference of squares (since it is equal to $(x^{2} - 1^{2}))$. Similarly, $(x^{2} - 9)$ is also a difference of squares since it can be written as $(x^{2} - 3^{2})$, and so it factors into $(x - 3)(x + 3)$.

So, we have $(x^{2} - 1)(x^{2} - 9) = (x - 1)(x + 1)(x - 3)(x + 3)$. Which values make this expression equal to 0? These are your roots or zeroes (roots and zeroes are two words for the same concept -- the numbers where your polynomial crosses the $x$-axis).

Now draw your $x$ and $y$-axis, and label your roots. Use test points between each of the roots to determine if the polynomial is positive or negative by plugging the test points into the polynomial. If the test point yields a positive number, then draw your polynomial from the right root on the $x$-axis upward then downward to connect to the left-most root. Repeat for each interval.

Answer 2

If you factor $(x^2-1)$ into $(x+1)$ and $(x-1)$ and factor $(x^2-9)$ into $(x+3)$ and $(x-3)$, you can determine that the $x$-intercepts for your equation are $1, -1, 3,$ and $-3$. Unless you need to graph it exactly, using the roots to do an approximation should suffice.

Answer 3

Hint: A good place to start is to find the intercepts. Note that $(x^2-y^2)=(x-y)(x+y)$