How do you graph `f(x)=x^4-4` using zeros and end behavior?

Graph `f(x)=x^4-4` using zeros and end behavior.

To find the zeros, factor the polynomial.

`f(x)=(x^2-2)(x^2+2)`

Factor again.

`f(x)=(x+sqrt2)^color(red)1(x-sqrt2)^color(red)1(x^2+2)`

Setting each factor equal to zero and solving gives:

`x=-sqrt2`, `x=sqrt2` and `x=+-sqrt2i`

The only real zeros are `sqrt2` and `-sqrt2`. Each has a multiplicity of `color(red)1` because the exponent on each factor is `color(red)1`. An odd multiplicity means the graph crosses (or cuts through) the `x` axis at the zeros/x-intercepts. The `x` intercepts are `(-sqrt2,0)` and `(sqrt2, 0)` which are approximately `(+-1.414,0)`.

To find the end behavior, examine the degree and leading coefficient of the original polynomial.

`f(x)=color(blue)1x^color(violet)4-4`

The degree is `color(violet)4` and the leading coefficient is `color(blue)1`.

An even degree with a positive leading coefficient indicates that as`xrarroo` and `xrarr-oo`, `f(x)rarroo`. In other words, the "ends" of the graph both point "up".

It is also helpful to find the `y` intercept. Setting `x=0` gives
`y=0^4-4=-4`. The `y` intercept is `(0, -4)`

The graph is shown below.