Question

Analyze each polynomial function.

$f\left(x\right)=x^4-18.5x^2+50.2619$

Short answer

The given function is:

$f\left(x\right)=x^4-18.5x^2+50.2619$

Step-by-step solution

Given Information

The given function is:

$f\left(x\right)=x^4-18.5x^2+50.2619$

The polynomial function

$f\left(x\right)=x^4-18.5x^2+50.2619$

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

The $y$-intercept is $f\left(0\right)=50.2619$and $x$-intercepts are $-3.9,-1.82,1.82$and $3.9.$

Values of x on each side of the x-intercepts:

$x$	$y$
$-4$	$10.26$
$-3$	$-35.24$
$-2$	$-7.74$
$-1$	$32.76$
$1$	$32.76$
$2$	$-7.74$
$3$	$-35.24$
$4$	$10.26$

The function has $3$turning points: $(-3.04,-35.3),(0,50.26)$and $(3.04,-35.3).$

Domain: $x\in \mathrm{ℝ}$Range: $<-35.3,\infty >$

The function is increasing on the intervals$(-3.04,0)$and $(3.04,\infty )$and decreasing on the interval $(-\infty ,-3.04)$and $(0,3.04).$

The graph