Expert: Sherman D. - 3/25/2006

Question
1)  What is the vertex and x-intercepts for the parabola whose equation is f(x)=x^2+4x+1?  State exact values, not decimal approximations.

2)  What is the remainder when 4x^3+6x^2-2x+5 is divided by x-1/2?

3)  Find all asymptotes for f(x)= 2x+4/x^2-4
vertical asymptote(s):____________________
horizontal asymptote(s):_____________________

4)  Find each of the following for the exponential function, f(x)=2^x-12 + 30
Domain:___________________
Range:_______________________

5.  The intensity of illumination on a surface varies inversely as the square of the distance of the light from the surface.  If the illumination from a source is 25 foot candles when the distance is 4 feet, then find the illumination when the distance is 6 feet.

Thanks!

Answer
1) What is the vertex and x-intercepts for the parabola whose equation is f(x)=x^2+4x+1? State exact values, not decimal approximations.

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

x = (-b)/2a
x = (-4)/2
x = -2

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

Vertex : (-2,-3)

x = (-b ± sqrt(b^2 - 4ac))/2a
x = (-4 ± sqrt((-4)^2 - 4(1)(1)))/(2(1))
x = (-4 ± sqrt(16 - 4))/2
x = (-4 ± sqrt(12))/2
x = (-4 ± sqrt(4 * 3))/2
x = (-4 ± 2sqrt(3))/2
x = -2 ± sqrt(3)

I cannot give you an exact value, because it is way to long, unless you meant to type f(x) = x^2 + 4x + 4

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If by this you mean

2)
(4x^3 + 6x^2 - 2x + 5)/((x - 1)/2)
((4x^3 + 6x^2 - 2x + 5)/1)/((x - 1)/2)
((4x^3 + 6x^2 - 2x + 5)/1)*(2/(x - 1))
(8x^3 + 12x^2 - 4x + 10)/(x - 1)

using synthetic division

1'|'8'12'-4'|'10
''|''''8'20'|'16
------------------
''|'8'20'16'|'26

so this becomes

8x^2 + 20x + 16 R 26

ANS : Remainder is 26

However if you meant (4x^3 + 6x^2 - 2x + 5)/(x - (1/2)), here goes

Using synthetic division

(1/2)'|'4'6'-2'|'5
''''''|'''2''4'|'1               
--------------------
''''''|'4'8''2'|'6

This gives you 4x^2 + 8x + 2 R 6

ANS : Remainder is 6

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

VA = -2 and 2
HA = 0

For a graph, go to www.calculator.com/calcs/GCalc.html
type in (2x + 4)/(x^2 - 4), and  you will see your VA and HA.

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4) Find each of the following for the exponential function, f(x) = 2^(x - 12) + 30

Range : y >= 30
Domain : x has + and - infinity

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5.)
y = k/x
or
k = xy

(25)(4)^2 = x(6)^2
25(16) = 36x
36x = 400
x = (400/36)
x = (100/9)
x = 11.1111111 feet