Question 1013093
Use the given function to answer the questions below

{{{f(x)=-x^4+x^2}}}

a. Use the leading coefficient test to determine the graphs end behavior

the leading coefficient of {{{a[n]x^n=-x^4}}} is {{{-1}}}

When {{{n}}} (which is {{{4}}} in your case) is {{{even}}} and {{{a[n]}}} is negative (which is {{{-1}}} in your case),graph {{{falls}}} to the left and right



b. Find the x-intercepts: 

{{{0=-x^4+x^2}}}
{{{x^4-x^2=0}}}
{{{(x^2-1)x^2=0}}}

if {{{x^2=0}}} we have double solution {{{x=0}}}
if {{{(x^2-1)=0}}}=>{{{x^2=1}}}=>{{{x=sqrt(1)}}}=>{{{x=1}}} or {{{x=-1}}}

so, the x-intercepts are at:
({{{0}}},{{{0}}})
({{{1}}},{{{0}}})
({{{-1}}},{{{0}}})



c. At which zeros does the graph of the function cross the x-axis:

the graph of the function cross the x-axis at {{{x=1}}} and {{{x=-1}}}


d. At which zero does the graph of the function touch the x-axis and turn around:

at {{{x=0}}} 


e. Find the y intercept buy computing f(0):

{{{f(0)=-0^4+0^2}}}
{{{f(0)=0}}}

so, the y-intercept is at:
({{{0}}},{{{0}}})

f. Determine the symmetry of the graph (odd, even, or neither):

Replace x with −x in order to determine if the function is odd, even, or neither.

{{{-(-x)^4+(-x)^2=-x^4+x^2}}}

The function is {{{even}}}.

y-axis is axis of symmetry


see it on a graph:


{{{ graph( 600, 600, -5, 5, -10, 5, -x^4+x^2) }}}