Question 963469: Part I.
G(x) = -x4 + 32x2 +144
a. Show whether G is an even, odd function or neither. If it’s even/odd, identify the property of its graph. (Must show your work)
(Help: To show that G is an even function, you must show that G(-x) = G(x), and G is an odd function you must show that G(-x) = -G(x), for ex. F(x) = x3 – 3x is an odd function, since:
F(-x) = (-x)3-3(-x) = -x3+3x = -(x3-3x) = -F(x), thus F is an odd function. And by definition odd function has graph symmetric about the origin).
b. There is a local maximum value of 400 at x = 4, determine the second local maximum value (Use property of the graph of an even/odd function from part a:
(Hints: if (x, f(x)) is a point on the graph of an even function f, then (-x, f(-x)) is symmetric about the y-axis since f(-x)= f(x) is also on the graph of f).
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.
(Hints: Use property of the graph of an even/odd function).
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).
(Hints: Y = |x + 4| + 7, then library function is f(x) = |x| absolute value function, then the graph of Y can be obtained from the graph of f by shifting the graph of f to the left 4 units, and up 7 units).
b. Use the library function f(x) =√x Write the function y obtained from f(x) =√x by shifting the graph of f 4 units to the left, and down 2 unit.
(Hints: f(x) = x2; f(x) = x3; f(x) = √x; f(x) = |x|, etc.. each is a library function. If y= f(x-h), then graph of y can be obtained from graph of f by shifting the graph of f to the right h units, etc…Please review transformations of graphs = shifting technique to answer both questions of part a, and b).
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website! Part I.
a.
A function is "even" when:
 for all 
A function is "odd" when:
 for all
check your function: is it "even"
f(x) = f(-x) for all x
 ...since  will be same as , and  will be same as  , we have
so, your function is 
is it odd:
for all
 .......your function is not 
b.
There is a local maximum value of  at  , determine the second local maximum value:
(, ) =( , )
find
so, if (, ) is a point on the graph of an even function G, then ( , ) is symmetric about the y-axis since  is also on the graph of G
c. Suppose the area under the graph of G between  , and  , that is bounded below by the x-axis is  square units, Using the result from part a to determine the area under the graph of G between  and , 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 and is bounded below by the x-axis is square units
Part II.
a.
Choose a library function, then use the library function, and describe how to graph the function:
 (just describe without graphing).
library function is 
and  can be obtained from the graph of  by shifting the graph of f to the right  units, and down  units
b.
Use the library function  Write the function y obtained from by shifting the graph of f units to the left, and down  unit.
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