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Math Review > Algebra > Flashcards

Algebra Flashcards

1
Q

Average rate of change

A

The average rate of change of function f over the interval a≤ x ≤b is given by this expression:
f(b) – f(a) / b – a

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2
Q

Greatest common monomial factor

A

Greatest integer and variable that is a factor of both numbers
10x^3 2 5 x x x
4x 2 2 x
GCF = 2x

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3
Q

Maximum amount of a quadratic function in a word problem

A

The x-coordinate of the vertex, which is the average of the two zeros. when it asks for maximum x this is answer?
Need to put the x-coordinate back into equation when it asks for maximum f of whatever. f(x)

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4
Q

Takeoff/liftoff height in quadratic function word problem

A

When x equals 0

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5
Q

km/h to m/s

A

x km/h * 1000m/1 km * 1 h/3600 s

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6
Q

Shifting absolute value graphs

A 
f(x) = a|x - h| + k 
a = how far graph stretches vertically, and whether it opens up or down 
h = how far graph shifts horizontally (- right(input +), + left (input -))
k = how far graph shifts vertically
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7
Q

Finding vertex (max/min) of a parabola (quadratic function)

A

For the x-value: -b/2a (ax^2 + bx + c)

For the y-value: plug x value into equation

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8
Q

degrees/min^2 to degrees/sec^2

A

x degrees/min^2 * 1 min^2/3600 sec^2 (60 * 60)

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9
Q

Shifting parabolas

A

(x - k)^2 + h
k = left/right - right + left
h = up/down

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10
Q

g(x) = a * b^x

A

a is the value of the function when x = 0 (y of y-intercept)
b is the constant ratio of g(x) for consecutive integer values of x
constant ratio: divide the y value of (1, y) by the y value of (0, y) (y-intercept)

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11
Q

g(r) = (r + n1)^2 + n2

A 
(r + n1)^2 + n2  = 0 
(r + n1)^2 = --n2
sqrt((r + n1)^2) = sqrt(--n2)
r + n1 = +/- of sqrt(--n2)
r = (+/- of sqrt(--n2)) -- n1 
zeros: + sqrt(--n2) -- n1  AND  --sqrt(--n2) -- n1
vertex: (--n1, n2)
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12
Q

point-slope formula

A

y - y1 = m(x - x1)
(x1, y1) = known point
m = slope
(x, y) = second point on the line, stays (x,y)?

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13
Q

slope formula

A

change in y / change in x

y2 - y1 / x2 - x1

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14
Q

explicit formula of geometric sequence

A

a(n) = k * r^(n-1)

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15
Q

recursive formula of geometric sequence

A 
a(1) = k 
a(n) = a(n-1) * r
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16
Q

number of solutions to linear equations

A 
1 = all EXCEPT 
0 = parallel lines, aka same slope 
infinite = same line
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17
Q

completing the square

A 
ax^2 + bx + c = 0 
x^2 + (b/a)x  + c/a = 0 
x^2 + (b/a)x =  --c/a 
OR
ax^2 + bx + c = 0 
a(x^2 + (b/a)x) + c = 0
d = completed square 
a(x^2 + (b/a)x) +d) + (c - (a*d))
e = completed square in (x - e)^2 form 
a(x - e)^2 + (c - (a*d))
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18
Q

find the nth term in the arithmetic sequence

A

(n-1) = the term before n, plug into equation to get n

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19
Q

recursive formula of arithmetic sequence

A 
a(1) = k 
a(n) = a(n-1) + r
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20
Q

explicit formula of arithmetic sequence

A

a(n) = k + r(n-1)

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21
Q

simplifying square roots

A
  1. factor the number under the radical
  2. when you find a perfect square, take it out and put it (once) in front of the radical
  3. do this until the number under the radical can’t be factored out (if more than one perfect squares in front of radicals, multiply them)
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22
Q

x times old problem

A

y number of years it will take to be x times as old
a1 + y a2 + y a = age
a1 + y = x(a2 + y)
simplify and solve for y

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23
Q

0^3

A

0

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24
Q

1^3

A

1

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25
Q

2^3

A

8

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26
Q

3^3

A

27

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27
Q

4^3

A

64

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28
Q

5^3

A

125

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29
Q

6^3

A

216

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30
Q

7^3

A

343

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31
Q

8^3

A

512

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32
Q

9^3

A

729

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33
Q

10^3

A

1000

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34
Q

11^3

A

1331

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35
Q

12^3

A

1728

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36
Q

13^3

A

2197

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37
Q

14^3

A

2744

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38
Q

15^3

A

3375

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39
Q

integer vs. real number

A

Integer – no fractions, counting (can be negative)

Real number – fractions

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40
Q

domain of a function

A

set of all real values of x that will give real values for y (input)

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41
Q

range of a function

A

set of all real values of y that you can get by plugging real numbers into x (output)

42
Q

zeros of (x+y)^2

A

–y

43
Q

zeros of (x+y)^2 – z

A 
(x+y)^2 = z 
√(x+y)^2 = √z
x+y = +/- √z
x = +/- √z -- y 
x = +√z -- y  AND -√z -- y
44
Q

exponential growth/decay

A

y = A * r^x
Increase/grow when common ratio is greater than one
Decrease/decay when common ratio is less than one
common ratio = r
y-intercept = A (x is equal to 0)

45
Q

discriminant of a quadratic function

A

b^2 – 4ac

46
Q

quadratic formula

A

-b +/- √b^2 – 4ac / 2a

47
Q

number of solutions according to discriminant

A

Positive discriminant = two distinct real number solutions (2 x-intercepts)
Negative discriminant = neither of solutions are real numbers (no x-intercept)
Discriminant of zero = repeated real number solution (1 x-intercept)

48
Q

quadratic equation

A

ax^2 + bx + c

49
Q

undefined slope

A

vertical line, all points on the line have the same x-coordinate, hence the denominator for slope formula would be zero, hence undefined
x = #

50
Q

slope of zero

A

horizontal line

y = # aka y = b

51
Q

is the graph a function

A

A graph represents a function if and only if every x-value on the graph corresponds to exactly one y-value.
To put it another way, if we can find an x-value with more than one corresponding y-value, then the graph doesn’t represent a function.
Vertical line test – does a vertical line draw intersect the graph only once
A horizontal line can be a function (put x in, get only one result), but a vertical line cannot (put in x, get infinite results)

52
Q

exponential growth equation

A

a(1+r)^x a = initial value r = growth or decay rate (most often represented as a percentage and expressed as a decimal ) x = the number of times the interval has passed

53
Q

exponential decay equation

A

a(1-r)^x a = initial value r = growth or decay rate (most often represented as a percentage and expressed as a decimal ) x = the number of times the interval has passed

54
Q

solving exponential equations with logarithms

A

a * b^x = d
b^x = d/a
logb(d/a) = x
log(d/a) / log(b) = x

55
Q

solving exponential equations with number/variable exponent logarithms

A

a * b^cx = d
b^cx = d/a
logb(d/a) = cx
(1/c)logb(d/a) = x

56
Q

solving square root equations (with radicals)

A
  1. Isolate the radical term
  2. Take square of both sides
  3. Check for extraneous solutions
57
Q

degrees to radians

A

x° * pi/180 (x degrees times (pi divided by 180))

58
Q

polynomial long division

A
  1. Divide the first term of the numerator by the first term of the denominator, and put that in the answer
  2. Multiply the denominator by that answer, put that below the numerator
  3. Subtract to create a new polynomial
    only divide xs by xs, but multiply by the whole thing
59
Q

rewrite exponential expressions as A*B^t

A 
(1/B)^x * (1/B)^x-y 
factor out (1/B)^x
(1/B)^x + (1/B)^x / (1/B)^y 
(1/B)^x + B^y * (1/B)^x 
(1/B)^x  + (1 + B^y) 
(1 + B^y) * (1/B)^x
60
Q

x^0

A

1

61
Q

x^-1

A

1/x

62
Q

x^m*x^n

A

x^m+n

63
Q

x^m/x^n

A

x^m-n

64
Q

(x^m)^n

A

x^mn

65
Q

(xy)^n

A

x^n*y^n

66
Q

(x-y)^n

A

x^n/y^n

67
Q

x^-n

A

1/x^n

68
Q

x^(m/n)

A

nroot(x^m)

(nroot(x))^m

69
Q

exponents of exponents

A

top first
x^y^z
y^z = a
x^a

70
Q

0^n

A

0

71
Q

0^-n

A

undefined (can’t divide by 0)

72
Q

+/- sqrt(-x)

A

+/- i * sqrt(x)
simplify sqrt(x) (y = squares outside)
+/-y*sqrt(x)i (i is outside radical)

73
Q

i

A

sqrt(-1)

74
Q

are the solutions to the discriminant real or complex

A

solutions are real if the discriminant (b^2 - 4ac) is ≥ 0

if less than zero, they are complex and i (square root of -1) is involved

75
Q

vertical compression/squash of graph

A

The expression k⋅f(x) when ∣k∣<1 is a vertical squash (or compression): The y-value of every point on the graph of y = f(x) is multiplied by k, so the points get closer to the x-axis

76
Q

stretch or compress graph

A

a*f(x)
a > 1 stretches it
0 < a < 1 compresses it

g(x) = ax2
if |a| > 1, the graph is stretched vertically , and if |a| <1 , the graph is compressed vertically
When a < 0, the parabola is then reflected across the x-axis

the graph of g is stretched vertically, so it will appear taller and narrower than the graph of f
the graph of g is compressed vertically, so it will appear shorter and wider than the graph of f

77
Q

A*B^f(t)

A 
A = initial quantity 
B = constant ratio (multiplied by) 
f(t) = an expression in terms of t that determines those time intervals   x days = t/x
78
Q

graphing logarithmic functions – determining the vertical asymptote

A
  1. Determine the vertical asymptote

- - the vertical asymptote of a logarithmic function occurs when the argument is equal to 0 (y) = 0

79
Q

parts of a logarithm

A 
log_x(y) = z 
x = base 
y = argument 
z = answer
80
Q

year to decade decay problem

A 
x = A * B^t 
10x = A * B^10t 
10x = A * (B^10)^t
81
Q

what does (x, y) on unit circle equal

A

(cos, sin)

82
Q

a * e^bx = d

A

e^bx = d/a
log_e(d/a) = bx
ln(d/a) = bx
ln(d/a) / b = x

83
Q

log_x(y)

A

x raised to what power equals y

84
Q

log_x(y) = z

A

x raised to z power equals y x^z = y

85
Q

change of base rule

A

log_b(a) = log_x(a) / log_x(b)

86
Q

quadratic equations with complex roots

A

if the discriminant (sqrt(b^2-4ac)) is negative, then i must be in the root, making it complex

87
Q

stretching/compressing graph in y direction

A

multiply/divide the output by a constant
vertex at (2,1)
2f(x) = (2, 3)
1/2f(x) = (2, 0.5)
each y-coordinate multiplied by the constant

88
Q

stretching/compression graph in x direction

A 
multiply/divide the input by a constant 
f(x) = (-2, 5) 
2f(x) = (-1, 5) 
1/2f(x) = (-4,5) 
each x-coordinate multiplied by the constant
89
Q

distance formula

A

distance = speed * time

90
Q

time formula

A

time = distance / speed

91
Q

delayed by time

A

delayed by half hour
time = k / v + 1/2
v = kilometers per hour

92
Q

average speed faster

A

traveling 20 kilo per hour faster
k / v + 20 > k / v
v = kilometers per horu

93
Q

second investment

A

put 100 dollars in at rate, invest again at same rate

100r*r = 100r^2

94
Q

x increased by y of its size

A

x * (1 + y )
x increased by 2/7 of its size = 1 + 2/7 = 9/7 = x * 9/7
x increased by

95
Q

graphing logarithmic functions – determining the x-intercept

A
  1. Determine the x-intercept
    – We can find the x-intercept of the graph by finding the x-value that makes y = 0
    — Divide both sides by first shift value (equals 0)
    — Rewrite so log subscript to the 0 power equals argument
    (will be 1 = argument)
    — Solve for x (x, 0)
    OR
    — Add/subtract last number to other side
    — Divide both sides by first shift value
    — Use answer as power for log subscript (1 = log3 to 3^1), set equal to x
    — Solve for x (x, 0)
96
Q

graphing logarithmic functions – finding another point on the graph

A
  1. Finding another point on the graph
    Choose x-values that result in argument equating to a power of the subscript of log?
    y = blog_c(x) — d
    log_cx = a (a for answer) === c^a= x
    y = b    a — d
    (x, y)
97
Q

exponents to logs

A

y = b^x = log_b(y) = x

98
Q

natural logarithm

A

log_e(x) = ln(x)

99
Q

e^y = x

A

then base e logarithm of x is

ln(x) = log_e(x) = y

100
Q

weekly to daily decay

A

A * B^(1/7)^x daily rate of decay = B^(1/7)

101
Q

quadratic patterns

A

(U + V)^2 = U^2 + 2UV + V^2
(U - V)^2 = U^2 – 2UV + V^2
(U+V)(U-V) = U^2 – V^2

Math Review (2 decks)

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