Algebra Flashcards
Average rate of change
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
Greatest common monomial factor
Greatest integer and variable that is a factor of both numbers
10x^3 2 5 x x x
4x 2 2 x
GCF = 2x
Maximum amount of a quadratic function in a word problem
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)
Takeoff/liftoff height in quadratic function word problem
When x equals 0
km/h to m/s
x km/h * 1000m/1 km * 1 h/3600 s
Shifting absolute value graphs
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
Finding vertex (max/min) of a parabola (quadratic function)
For the x-value: -b/2a (ax^2 + bx + c)
For the y-value: plug x value into equation
degrees/min^2 to degrees/sec^2
x degrees/min^2 * 1 min^2/3600 sec^2 (60 * 60)
Shifting parabolas
(x - k)^2 + h
k = left/right - right + left
h = up/down
g(x) = a * b^x
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)
g(r) = (r + n1)^2 + n2
(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)
point-slope formula
y - y1 = m(x - x1)
(x1, y1) = known point
m = slope
(x, y) = second point on the line, stays (x,y)?
slope formula
change in y / change in x
y2 - y1 / x2 - x1
explicit formula of geometric sequence
a(n) = k * r^(n-1)
recursive formula of geometric sequence
a(1) = k a(n) = a(n-1) * r
number of solutions to linear equations
1 = all EXCEPT 0 = parallel lines, aka same slope infinite = same line
completing the square
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))
find the nth term in the arithmetic sequence
(n-1) = the term before n, plug into equation to get n
recursive formula of arithmetic sequence
a(1) = k a(n) = a(n-1) + r
explicit formula of arithmetic sequence
a(n) = k + r(n-1)
simplifying square roots
- 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)
x times old problem
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
0^3
0
1^3
1
2^3
8
3^3
27
4^3
64
5^3
125
6^3
216
7^3
343
8^3
512
9^3
729
10^3
1000
11^3
1331
12^3
1728
13^3
2197
14^3
2744
15^3
3375
integer vs. real number
Integer – no fractions, counting (can be negative)
Real number – fractions
domain of a function
set of all real values of x that will give real values for y (input)
range of a function
set of all real values of y that you can get by plugging real numbers into x (output)
zeros of (x+y)^2
–y
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
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)
discriminant of a quadratic function
b^2 – 4ac
quadratic formula
-b +/- √b^2 – 4ac / 2a
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)
quadratic equation
ax^2 + bx + c
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 = #
slope of zero
horizontal line
y = # aka y = b
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)
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
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
solving exponential equations with logarithms
a * b^x = d
b^x = d/a
logb(d/a) = x
log(d/a) / log(b) = x
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
solving square root equations (with radicals)
- Isolate the radical term
- Take square of both sides
- Check for extraneous solutions
degrees to radians
x° * pi/180 (x degrees times (pi divided by 180))
polynomial long division
- Divide the first term of the numerator by the first term of the denominator, and put that in the answer
- Multiply the denominator by that answer, put that below the numerator
- Subtract to create a new polynomial
only divide xs by xs, but multiply by the whole thing
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
x^0
1
x^-1
1/x
x^m*x^n
x^m+n
x^m/x^n
x^m-n
(x^m)^n
x^mn
(xy)^n
x^n*y^n
(x-y)^n
x^n/y^n
x^-n
1/x^n
x^(m/n)
nroot(x^m)
(nroot(x))^m
exponents of exponents
top first
x^y^z
y^z = a
x^a
0^n
0
0^-n
undefined (can’t divide by 0)
+/- sqrt(-x)
+/- i * sqrt(x)
simplify sqrt(x) (y = squares outside)
+/-y*sqrt(x)i (i is outside radical)
i
sqrt(-1)
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
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
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
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
graphing logarithmic functions – determining the vertical asymptote
- Determine the vertical asymptote
- - the vertical asymptote of a logarithmic function occurs when the argument is equal to 0 (y) = 0
parts of a logarithm
log_x(y) = z x = base y = argument z = answer
year to decade decay problem
x = A * B^t 10x = A * B^10t 10x = A * (B^10)^t
what does (x, y) on unit circle equal
(cos, sin)
a * e^bx = d
e^bx = d/a
log_e(d/a) = bx
ln(d/a) = bx
ln(d/a) / b = x
log_x(y)
x raised to what power equals y
log_x(y) = z
x raised to z power equals y x^z = y
change of base rule
log_b(a) = log_x(a) / log_x(b)
quadratic equations with complex roots
if the discriminant (sqrt(b^2-4ac)) is negative, then i must be in the root, making it complex
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
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
distance formula
distance = speed * time
time formula
time = distance / speed
delayed by time
delayed by half hour
time = k / v + 1/2
v = kilometers per hour
average speed faster
traveling 20 kilo per hour faster
k / v + 20 > k / v
v = kilometers per horu
second investment
put 100 dollars in at rate, invest again at same rate
100r*r = 100r^2
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
graphing logarithmic functions – determining the x-intercept
- 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)
graphing logarithmic functions – finding another point on the graph
- 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 = ba — d
(x, y)
exponents to logs
y = b^x = log_b(y) = x
natural logarithm
log_e(x) = ln(x)
e^y = x
then base e logarithm of x is
ln(x) = log_e(x) = y
weekly to daily decay
A * B^(1/7)^x daily rate of decay = B^(1/7)
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
