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
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
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)
4
Q
Takeoff/liftoff height in quadratic function word problem
A
When x equals 0
5
Q
km/h to m/s
A
x km/h * 1000m/1 km * 1 h/3600 s
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
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
8
Q
degrees/min^2 to degrees/sec^2
A
x degrees/min^2 * 1 min^2/3600 sec^2 (60 * 60)
9
Q
Shifting parabolas
A
(x - k)^2 + h
k = left/right - right + left
h = up/down
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)
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)
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)?
13
Q
slope formula
A
change in y / change in x
y2 - y1 / x2 - x1
14
Q
explicit formula of geometric sequence
A
a(n) = k * r^(n-1)
15
Q
recursive formula of geometric sequence
A
a(1) = k a(n) = a(n-1) * r
16
Q
number of solutions to linear equations
A
1 = all EXCEPT 0 = parallel lines, aka same slope infinite = same line
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))
18
Q
find the nth term in the arithmetic sequence
A
(n-1) = the term before n, plug into equation to get n
19
Q
recursive formula of arithmetic sequence
A
a(1) = k a(n) = a(n-1) + r
20
Q
explicit formula of arithmetic sequence
A
a(n) = k + r(n-1)
21
Q
simplifying square roots
A
- factor the number under the radical
- when you find a perfect square, take it out and put it (once) in front of the radical
- 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)
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
23
Q
0^3
A
0
24
Q
1^3
A
1
25
2^3
8
26
3^3
27
27
4^3
64
28
5^3
125
29
6^3
216
30
7^3
343
31
8^3
512
32
9^3
729
33
10^3
1000
34
11^3
1331
35
12^3
1728
36
13^3
2197
37
14^3
2744
38
15^3
3375
39
integer vs. real number
Integer -- no fractions, counting (can be negative)
| Real number -- fractions
40
domain of a function
set of all real values of x that will give real values for y (input)
41
range of a function
set of all real values of y that you can get by plugging real numbers into x (output)
42
zeros of (x+y)^2
--y
43
zeros of (x+y)^2 -- z
```
(x+y)^2 = z
√(x+y)^2 = √z
x+y = +/- √z
x = +/- √z -- y
x = +√z -- y AND -√z -- y
```
44
exponential growth/decay
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
discriminant of a quadratic function
b^2 -- 4ac
46
quadratic formula
-b +/- √b^2 -- 4ac / 2a
47
number of solutions according to discriminant
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
quadratic equation
ax^2 + bx + c
49
undefined slope
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
slope of zero
horizontal line
| y = # aka y = b
51
is the graph a function
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
exponential growth equation
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
exponential decay equation
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
solving exponential equations with logarithms
a * b^x = d
b^x = d/a
logb(d/a) = x
log(d/a) / log(b) = x
55
solving exponential equations with number/variable exponent logarithms
a * b^cx = d
b^cx = d/a
logb(d/a) = cx
(1/c)logb(d/a) = x
56
solving square root equations (with radicals)
1. Isolate the radical term
2. Take square of both sides
3. Check for extraneous solutions
57
degrees to radians
x° * pi/180 (x degrees times (pi divided by 180))
58
polynomial long division
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
rewrite exponential expressions as A*B^t
```
(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
x^0
1
61
x^-1
1/x
62
x^m*x^n
x^m+n
63
x^m/x^n
x^m-n
64
(x^m)^n
x^mn
65
(xy)^n
x^n*y^n
66
(x-y)^n
x^n/y^n
67
x^-n
1/x^n
68
x^(m/n)
nroot(x^m)
| (nroot(x))^m
69
exponents of exponents
top first
x^y^z
y^z = a
x^a
70
0^n
0
71
0^-n
undefined (can't divide by 0)
72
+/- sqrt(-x)
+/- i * sqrt(x)
simplify sqrt(x) (y = squares outside)
+/-y*sqrt(x)i (i is outside radical)
73
i
sqrt(-1)
74
are the solutions to the discriminant real or complex
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
vertical compression/squash of graph
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
stretch or compress graph
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
A*B^f(t)
```
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
graphing logarithmic functions -- determining the vertical asymptote
1. Determine the vertical asymptote
| - - the vertical asymptote of a logarithmic function occurs when the argument is equal to 0 (y) = 0
79
parts of a logarithm
```
log_x(y) = z
x = base
y = argument
z = answer
```
80
year to decade decay problem
```
x = A * B^t
10x = A * B^10t
10x = A * (B^10)^t
```
81
what does (x, y) on unit circle equal
(cos, sin)
82
a * e^bx = d
e^bx = d/a
log_e(d/a) = bx
ln(d/a) = bx
ln(d/a) / b = x
83
log_x(y)
x raised to what power equals y
84
log_x(y) = z
x raised to z power equals y x^z = y
85
change of base rule
log_b(a) = log_x(a) / log_x(b)
86
quadratic equations with complex roots
if the discriminant (sqrt(b^2-4ac)) is negative, then i must be in the root, making it complex
87
stretching/compressing graph in y direction
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
stretching/compression graph in x direction
```
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
distance formula
distance = speed * time
90
time formula
time = distance / speed
91
delayed by time
delayed by half hour
time = k / v + 1/2
v = kilometers per hour
92
average speed faster
traveling 20 kilo per hour faster
k / v + 20 > k / v
v = kilometers per horu
93
second investment
put 100 dollars in at rate, invest again at same rate
| 100r*r = 100r^2
94
x increased by y of its size
x * (1 + y )
x increased by 2/7 of its size = 1 + 2/7 = 9/7 = x * 9/7
x increased by
95
graphing logarithmic functions -- determining the x-intercept
2. 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
graphing logarithmic functions -- finding another point on the graph
3. Finding another point on the graph
Choose x-values that result in argument equating to a power of the subscript of log?
y = b*log_c(x) — d
log_cx = a (a for answer) === c^a= x
y = b*a — d
(x, y)
97
exponents to logs
y = b^x = log_b(y) = x
98
natural logarithm
log_e(x) = ln(x)
99
e^y = x
then base e logarithm of x is
| ln(x) = log_e(x) = y
100
weekly to daily decay
A * B^(1/7)^x daily rate of decay = B^(1/7)
101
quadratic patterns
(U + V)^2 = U^2 + 2UV + V^2
(U - V)^2 = U^2 -- 2UV + V^2
(U+V)(U-V) = U^2 -- V^2
