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factors of -15 that "combine" to -2

like 3 and -5

(a+3b)(a-5b)

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difference of squares __ (p^2+1)(p^2-1)

difference of squares (again) __ (p^2+1)(p+1)(p-1)

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difference of squares __ (p^2+1)(p^2-1)

difference of squares (again) __ (p^2+1)(p+1)(p-1)

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factors of 9 that can "combine" with factors of 4 to get -12

like -3 and 2 __ (2c-3)(2c-3)

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find the z values for the upper and bounds of the range and then find the portion of the distribution represented

lower __ z=(550.67-636.55)/89.46 __ z=-.96 (approx)

upper __ z=(836.94-636.55)/89.46 __ z=2.24 (approx)

this range represents about 82% of the distribution

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since the z value is negative, it is to the left of the mean

this means that everything to the right of the value is more than half of the area

only one answer is more than .5

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plotting the graphs is straight forward (a graphing calculator would be a big help)
__ the f(x) values are on the vertical (y) axis and the x values are on the horizontal axis
__ find the f(x) values by "plugging in" values for x

1. f(x)=7^x __ when x=0, f(x)=1 (this is the y-intercept) __ when x is 1, f(x)=7
__ as x becomes a large NEGATIVE value, f(x) approaches zero (horizontal asymptote)

2. f(x)=4^(x-3) __ when x=3, the exponent is 0 so f(x)=1 __ when x=0, f(x)=4^(-3) or 1/64
__ same general shape as #1 with different y-intercept

3. f(x)=(1/5)^x __ when x=0, f(x)=1
__ as x becomes a large POSITIVE value, f(x) approaches zero (horizontal asymptote)
__ this graph is sort of a "mirror image" of #'s 1 and 2

4. logarithms are NOT defined for negative quantities, so this graph is only on the right-hand side of the vertical axis
__ as x approaches zero (very small fractions), f(x) approaches negative infinity (vertical asymptote)

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$$$, huh...WOW!!

plotting the graphs is straight forward (a graphing calculator would be a big help)
__ the f(x) values are on the vertical (y) axis and the x values are on the horizontal axis
__ find the f(x) values by "plugging in" values for x

1. f(x)=7^x __ when x=0, f(x)=1 (this is the y-intercept) __ when x is 1, f(x)=7
__ as x becomes a large NEGATIVE value, f(x) approaches zero (horizontal asymptote)

2. f(x)=4^(x-3) __ when x=3, the exponent is 0 so f(x)=1 __ when x=0, f(x)=4^(-3) or 1/64
__ same general shape as #1 with different y-intercept

3. f(x)=(1/5)^x __ when x=0, f(x)=1
__ as x becomes a large POSITIVE value, f(x) approaches zero (horizontal asymptote)
__ this graph is sort of a "mirror image" of #'s 1 and 2

4. logarithms are NOT defined for negative quantities, so this graph is only on the right-hand side of the vertical axis
__ as x approaches zero (very small fractions), f(x) approaches negative infinity (vertical asymptote)

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a formula is not needed __ this is a proportional problem __ k is the constant of proportionality

3600=[k*1800*20^2]/600 __ 3600=k*1200 __ dividing by 1200 __ 3=k

f=[3*1800*50^2]/570 __ f=23684

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because of gravitational acceleration, r is constantly changing __ that makes d=rt not useful in this situation

when it is on the ground, h=0 __ so we can substitute and solve

0=64t-16t^2 __ factoring __ 0=(16t)(4-t)

0=16t __ dividing by 16 __ 0=t __ this is the beginning of the "flight"

0=4-t __ adding t __ t=4 __ this is the "flight" time

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a general trajectory related formula for baseballs, golf balls, artillery shells, etc.
__ is h=-16t^2+(Vo)t+Ho __ h is the height at time (t), Vo is the initial upward (vertical) velocity
__ and Ho is the initial height

for part a), you need to find the time (t) when h=0 (return to Earth)

0=-16t^2+150t+5 __ use the quadratic formula to find t
__ HINT __ the positive value is the one you want

other HINT __ max height is approx 357'

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you found the upper limit; what about the lower limit?

there is a tolerance "band" __ lower to 4914

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with complex numbers, the real part is on the horizontal (x) axis
__ and the imaginary part is on the vertical (y) axis

so sqrt(3) on the x and 1 on the y forms two sides of a right triangle
__ the 3rd side is the hypotenuse; which is the polar radius

using Pythagoras, r^2=x^2+y^2 __ r^2=[sqrt(3)]^2+1^2 __ r^2=3+1 __ r=2

the tangent of the polar angle is [imaginary]/[real] or y/x
__ tan(Θ)=1/sqrt(3) __ Θ=30º or π/6 radians

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"If the total cost of x apples is b cents"; then each apple costs b/x

so, y apples will cost y*(b/x) or yb/x

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(2x-3y)(x+2y)

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factoring __

cancelling __

"cross" multiplying __ x^2-9=x^+5x+4 __ subtracting x^2+4 __ -13=5x __ dividing by -13 __ x=-5/13

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let t=time in months __ a=2p (doubling) __ r=.05

a=p(1+(r/12))^t __ dividing by p __ 2=(1+(.05/12)^t __ taking log __ log(2)=t*log(1+(.05/12)

dividing by log(1+(.05/12) __ [log(2)]/[log(1+(.05/12)]=t

t=166.7 (approx) __ should be 167 to get final compounding

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demonstration is the most straight forward

using rules for exponents, (x^a)/(x^b)=x^(a-b) __ (x^5)/(x^3)=x^2 __ (x*x*x*x*x)/(x*x*x)=x*x

suppose a=b __ (x^a)/(x^b)=x^(a-b)=x^0 __ (x*x*x)/(x*x*x)=1=x^0

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the number of ways to select 4 funds out of 10 is 10C4 or 210

the number of ways to select 3 funds out of 6 is 6C3 or 20

the number of ways to have 3 good funds with 1 poor fund is 20*6 or 120

so the probability of 3 out of 4 funds being good is 120/210 or 4/7

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15 years, huh...congratulations

for an expression of the form ax^2+bx+c, the equation for the axis of symmetry is x=-b/(2a)

since the vertex lies on the axis of symmetry,
__ substituting this x value into the equation of the function will give the y value (and thus the coordinates) of the vertex

the intercepts, where the graph crosses an axis, are found by substituting zero for x or y and solving for the other
__ this is because when you are crossing an axis, the value of the other component is zero

axis of symmetry __ x=-3/(2*1) __ x=-3/2

vertex __ y=(-3/2)^2+3(-3/2)-10 __ y=9/4-9/2-10 __ y=-49/4
__ so (-3/2,-49/4) is the location of the vertex

intercepts __ substituting 0 for x __ y=0^2+3(0)-10 __ y=-10 __ this is the y intercept

substituting 0 for y __ 0=x^2+3x-10 __ factoring __ 0=(x+5)(x-2)
x+5=0 __ x=-5
x-2=0 __ x=2
these are the x intercepts (the axis of symmetry is midway between them)

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the radical in the denominator is a no-no
__ so it is removed by multiplying the numerator and denominator by 5+sqrt(3), the conjugate of the denominator

2[5+sqrt(3)]/[(5-sqrt(3))*(5+sqrt(3))] __ [10+2sqrt(3)]/[5^2-(sqrt(3))^2] __ [10+2sqrt(3)]/(25-3)

[5+sqrt(3)]/11

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common factors first __ 2x(x^2-3)=0

factoring the remaining quadratic (difference of squares) __ 2x(x+sqrt(3))(x-sqrt(3))=0

2x=0 __ x=0

x+sqrt(3)=0 __ x=-sqrt(3)

x-sqrt(3)=0 __ x=sqrt(3)

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1. for f(x)=3^xn __ at x=0, f(x)=1 __ as x gets more negative, f(x) approaches zero (negative x-axis is horizontal asymptote)
__ no asymptote for positive x

for g(x)=log52x __ as x approaches zero, g(x) approaches -∞ (negative y-axis is vertical asymptote)
__ logarithms are not defined for negative arguments (negative numbers don't have logs)
__ no asymptote for positive x

2. there are 45 (180/4) 4 min periods in 3 hr __ the cells are doubling every 4 min
__ the number of cells after 3 hrs would be 2^45 __ 35184372088832 (calculators are useful)

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multiplying the numerator and denominator by sin(A)*cos(A) gives [sin^2(A)-cos^2(A)]/[sin^2(A)+cos^2(A)]

since sin^2(A)+cos^2(A)=1, we are left with sin^2(A)-cos^2(A) or 1-2cos^2(A)

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the angle whose tangent is -sqrt(3) is part of a 30º-60º-90º triangle in the 2nd quadrant (tangent is negative)

the reference angle in the 2nd quadrant is 60º, so the angle is 120º or 2pi/3 radians

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subtracting 8T+28 __ 0=T^2-T-28

using quadratic formula __ T=[1±sqrt(1-(4*1*(-28)))]/2 __ T=[1±sqrt(113)]/2

T=5.82 (approx)

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there are 20C3 (1140) possible groups of 3 in 20 __ some of these groups contain non-purchasers

there are 18C3 (816) possible groups of 3 in 18 __ none of these groups contain non-purchasers

probability of all purchasers is 816/1140 or 0.716 (approx)

if order of selection is thought to be important (permutation vs combination); the result will be the same

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there are 20C3 (1140) possible groups of 3 in 20 __ some of these groups contain non-purchasers

there are 18C3 (816) possible groups of 3 in 18 __ none of these groups contain non-purchasers

probability of all purchasers is 816/1140 or 0.716 (approx)

if order of selection is thought to be important (permutation vs combination); the result will be the same

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the two smaller triangles created by the altitude are similar to the original triangle (and each other)

using the similarity ratios, the altitude is the geometric mean of the two segments

8/a=a/18 __ "cross" multiplying __ a^2=144 __ taking square root __ a=12

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good so far __ be careful with parentheses - to many is better than not enough

M=3/[3-(-2/3)] __ M=3/(11/3) __ M=9/11

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for simple interest __ I=P*R*T

dividing by R*T __ I/(R*T)=P __ 245/(.035*3.5)=P __ 2000=P