Question

Analyze each polynomial function.

$f\left(x\right)=x^4-2.5x^2+0.5625$

Short answer

The given function is:

$f\left(x\right)=x^4-2.5x^2+0.5625$

Step-by-step solution

Given Information

The given function is:

$f\left(x\right)=x^4-2.5x^2+0.5625$

The polynomial function

$f\left(x\right)=x^4-2.5x^2+0.5625$

The polynomial function $f$is of degree $4$and the graph behaves like $y=x^4.$

The $y$-intercept is $f\left(0\right)=0.5625$and $x$-intercepts are $-1.5,-0.5,0.5$and $1.5.$

Values x on each side of the x-intercepts:

$x$	$y$
$-2$	$6.56$
$-1$	$-0.94$
$0$	$0.56$
$1$	$-0.94$
$2$	$6.56$

The function has $3$turning points: $(-1.12,-1),(0,0.56)$and $(1.12,-1).$

Domain: $x\in \mathrm{ℝ}$Range:$<-1,\infty >$

The function is increasing on the intervals $(-1.12,0)$and$(1.12,\infty )$and decreasing on the intervals $(-\infty ,-1.12)$and$(0,1.12).$

The graph