Question 54753
Prove Analytically that f(x)=x^3-4x is odd.
Substitute -x in for x, if f(-x)=-f(x) then the function is odd.
{{{f(-x)=(-x)^3-4(-x)}}} remember (-1)^(odd#)=-1
{{{f(-x)=-x^3+4x}}}
Therefore f(-x)=-f(x), so the function is odd.
:
Prove Analytically the f(x)=x^4+2x^2+5 is even.
Substitute -x in for x, if f(-x)=f(x) then the function is even.
{{{f(-x)=(-x)^4+3(-x)^2+5}}} remember (-1)^(even#)=1
{{{f(-x)=x^4+3x^2+5}}}
Therefore f(-x)=f(x), so the function is even.
:
Write the equation for a function that has a graph with the shape of y=square root of x, but is reflected in the y-axis and shifted up 12.
f(-x) results in a reflection about the y-axis.
f(x)+k recults in a vertical shift k units.
{{{y=sqrt(x)}}} Looks like:
{{{graph(300,200,-10,10,-10,10,sqrt(x))}}}
{{{y=sqrt(-x)}}} reflects about y-axis.
{{{y=sqrt(-x)+12}}} reflects it about y-axis and shiftes the graph up 12 units.
It looks like:
{{{graph(300,200,-10,10,-5,15,sqrt(-x)+12)}}}
:
Write the equation for a function that has a graph of y=x^2, but is vertically shrunk by a factor of 0.2 and shifted left 15 units.
af(x) results in a vertical shrink if -1<x<1
f(x+k) results in a horizontal shift left of k units.
{{{y=x^2}}}  Looks like:
{{{graph(300,200,-10,10,-10,10,x^2)}}}
{{{y=.2x^2}}} results in a vertical shink of a factor of .2
{{{y=.2(x+15)^2}}} results in the shrink and the left shift of 15 units.
It looks like:
{{{graph(300,200,-20,5,-10,10,.2(x+15)^2)}}}
Happy Calculating!!!