Question 963469
Part I. 

a.

A function is "even" when:

{{{f(x) = f(-x)}}} for all {{{x}}}

A function is "odd" when:

{{{-f(x) = f(-x)}}} for all {{{x}}}

check your function:  is it "even"
f(x) = f(-x) for all x
{{{G(-x)  =  -(-x)^4+32(-x)^2+144}}}...since {{{(-x)^4}}} will be same as{{{ (x)^4}}}, and  {{{(-x)^2}}} will be same as {{{(x)^2}}}, we have

{{{G(-x)  =  -x^4+32x^2+144}}}

so, your function is {{{even}}}

is it odd:

{{{-f(x) = f(-x)}}} for all {{{x}}}
{{{ -( -x^4+32x^2+144)=-(-x)^4+32(-x)^2+144}}}

{{{ x^4-32x^2-144<>-x^4+32x^2+144}}}.......your function is not {{{odd}}}


b.
There is a local maximum value of {{{400}}} at {{{x = 4}}}, determine the second local maximum value:

({{{x}}},{{{ f(x)}}}) =({{{4}}}, {{{400}}})
find
{{{G(-4)  =  -(-4)^4+32(-4)^2+144}}}
{{{G(-4)  =  -256+32*16+144}}}
{{{G(-4)  =  -256+512+144}}}
{{{G(-4)  = 400}}}

 so, if ({{{4}}}, {{{400}}}) is a point on the graph of an even function G, then ({{{-4}}}, {{{400}}})  is symmetric about the y-axis since {{{G(-x)= G(x)}}} is also on the graph of G


c. Suppose the area under the graph of G between {{{x = 0}}}, and {{{x = 6}}}, that is bounded below by the x-axis is {{{1612.8}}} square units, Using the result from part a to determine the area under the graph of G between {{{x = -6}}} and {{{x = 0}}}, bounded below by the x-axis.



Graphs of even and odd functions have following properties:

    If function is even then its graph is symmetric about y-axis.
    If function is odd then its graph is symmetric about origin.

so, because the  graph is symmetric about y-axis,  the area under the graph of G between {{{x = -6}}} and {{{x = 0}}}  is bounded below by the x-axis is {{{1612.8}}} square units


Part II. 

a.

Choose a library function, then use the library function, and describe how to graph the function:
{{{Y = (x-3)^2 -5}}} (just describe without graphing).

library function is {{{f(x) =x^2}}}
and {{{Y}}} can be obtained from the graph of {{{f(x)}}} by shifting the graph of f to the right {{{3}}} units, and down {{{5}}} units

 b.

 Use the library function {{{f(x) =sqrt(x)}}} Write the function y obtained from {{{f(x) =sqrt(x)}}} by shifting the graph of f {{{4}}} units to the left, and down {{{2}}} unit.

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

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