Expert: Sherman D. - 4/22/2006

Question
1.  Find each of the following for f(x)=3(x+4)^2-30. (No decimal approximations, find exact values).

a)  vertex:______________________
b)  asix of symmetry:________________________
c)  y-intercept(s):____________________
d)  x-intercept(s):


2)  Find the vertex and intercepts.  Use these to graph the quadratic function.
f(x)=2x^2+4x-6.

3)  The function f(x)= -x^2+42x-360 models the daily profit (f(x) in hundreds of dollars) for a company that manufactures x amount of VCR's daily.  How many VCR's should be manufactured each day to maximize their profit?  What is the maximum daily profit?

4)  Consider the function f(x)= -x^4 + 2x^3 -x^2.

a)  Use Factoring to find all the zeros of f.

b)  Use the Leading Coefficient Test and the zeros of f to sketch the graph of the function.

5)  Is x+1 is a factor or
f(x)=4x^4-x^3-3x^2+2x-2?  Why or why not?


6)  Suppose we know that 3 is a zero of f(x)=2x^3-3x^2-11x+6.  Use synthetic division to find the zeros of f.

7)  Write and equation which has 0, -3, and i as zeros.  (hint:  leave your equation in factored form).

Thanks,
Dan

Answer
1.) Find each of the following for f(x)=3(x+4)^2-30. (No decimal approximations, find exact values). If you want exact values, you will have to plug in the values on a calculator.

f(x) = 3(x + 4)^2 - 30
f(x) = 3((x + 4)^2 - 10)
f(x) = 3(((x + 4)(x + 4)) - 10)
f(x) = 3((x^2 + 8x + 16) - 10)
f(x) = 3(x^2 + 8x + 16 - 10)
f(x) = 3(x^2 + 8x + 6) or 3x^2 + 24x + 18

using the quadratic formula
x = (-8 ± sqrt(64 - 24))/2
x = (-8 ± sqrt(40))/2
x = (-8 ± 2sqrt(10))/2
x = -4 ± sqrt(10)

x = -b/2a
x = -8/2
x = -4

f(-4) = 3((-4)^2 + 8(-4) + 6)
f(-4) = 3(16 - 32 + 6)
f(-4) = 3(-16 + 6)
f(-4) = 3(-10)
f(-4) = -30

Vertex : (-4,-30)

a.) Vertex : (-4,-30)
b.) AOS : x = -4
c.) y-intercept : y = 18
d.) x-intercepts : -4 + sqrt(10) and -4 - sqrt(10)

For a graph, go to www.calculator.com/calcs/GCalc.html

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2) Find the vertex and intercepts. Use these to graph the quadratic function.

f(x) = 2x^2 + 4x - 6
f(x) = 2(x^2 + 2x - 3)
f(x) = 2(x + 3)(x - 1)
x = -3 or 1

x = -b/2a
x = -4/4
x = -1

f(-1) = 2((-1)^2 + 2(-1) - 3)
f(-1) = 2(1 - 2 - 3)
f(-1) = 2(-1 - 3)
f(-1) = 2(-4)
f(-1) = -8

Vertex = (-1,-8)

Vertex : (-1,-8)
x-intercepts : -3 and 1
y-intercept : -6

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3) The function f(x)= -x^2 + 42x - 360 models the daily profit (f(x) in hundreds of dollars) for a company that manufactures x amount of VCR's daily. How many VCR's should be manufactured each day to maximize their profit? What is the maximum daily profit?

x = -b/2a
x = -42/-2
x = 21

f(21) = -(21)^2 + 42(21) - 360
f(21) = -441 + 882 - 360
f(21) = 441 - 360
f(21) = 81

21 VCRs to make a maximum profit of $8,100

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4) Consider the function f(x)= -x^4 + 2x^3 - x^2.

f(x) = -x^4 + 2x^3 - x^2
f(x) = (-x^2)(x^2 - 2x + 1)
f(x) = (-x^2)(x - 1)(x - 1)

a) x = 0 and 1
b) For a graph, go to the site mentioned above.

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5) Is x+1 is a factor or
f(x) = 4x^4 - x^3 - 3x^2 + 2x - 2? Why or why not?

Using synthetic division

-1'|'4''-1'-3''2'|'-2
'''|''''-4''5'-2'|'-1
-------------------------
'''|'4''-5''2''0'|'-3

4x^3 - 5x^2 + 2x R -3

Since this problem has a remainder, x + 1 is not a factor of f(x) = 4x^4 - x^3 - 3x^2 + 2x - 2

If you go to www.quickmath.com, click on factor and type in 4x^4 - x^3 - 3x^2 + 2x - 2, you will find out that x + 1 isn't a factor, but x - 1 is.

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6) Suppose we know that 3 is a zero of f(x)=2x^3-3x^2-11x+6. Use synthetic division to find the zeros of f.

3'|'2'-3'-11'|''6
''|''''6'''9'|'-6
-------------------
''|'2''3''-2'|''0

2x^2 + 3x - 2 = (x + 2)(2x - 1)

The other 2 zeros are -2 and (1/2)

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7) Write and equation which has 0, -3, and i as zeros. (hint: leave your equation in factored form).

x(x + 3)(x - i)(x + i)
x(x + 3)(x^2 + ix - ix - (i)^2)
x(x + 3)(x^2 - (sqrt(-1))^2)
x(x + 3)(x^2 - (-1))
(x^2 + 3x)(x^2 + 1)
x^4 + x^2 + 3x^3 + 3x
x^4 + 3x^3 + x^2 + 3x

If you go to www.quickmath.com, click on simplify, and type in x(x + 3)(x - i)(x + i) it will simplify it to x(x^3 + 3x^2 + x + 3), which is the same thing as my answer.