Math 7 - 12 TEXES Content Exam
derivative of a^u
(a^u)(u')(lna)
Rotate 180 degrees about the origin
(x,y) -> (-x,-y)
Rotate 90 degrees counterclockwise
(x,y) -> (-y,x)
Rotate 270 degrees counterclockwise
(x,y) -> (y, -x)
derivative of cosx
-sinx
standard deviation (steps)
1. Calculate the mean 2. Calculate each variable (x) minus the mean. (x-x̄) 3. Square each value left over 4. Add them all together 5. Divide by n 6. Take the square root
How to calculate variance
1. find mean 2. Subtract each score from the mean 2. Square every deviation of the mean 3. Find mean again
What makes a function not differentiable?
1. has a cusp 2. has a hole 3. limit is not the same approaching both sides 4. not a funct (vert line test)
Calculate Regression line in calc
1. stat - edit 2. enter lists in L1 and L2 3. 2nd - y= - plot 1 ON - L1, L2 5) stat - calc - linreg - enter
binary system notation
2^n
Empirical Rule (stats)
68%, 95%, 99.7%
response variable
A variable that measures an outcome of a study. (dependent)
Area of equilateral triangle
A=s² (√3)/4
inverse
If not p, then not q
Contrapositive
If not q, then not p
converse
If q, then p
How to make the center of a circle
find the perpendicular bisectors of two chords
deductive reasoning
general to specific
Is a matrix invertible?
if ad - bc = 0
odd function
left half is reflected and flipped upside down perfectly! graph is symmetrical with respect to the origin; f(-x)=-f(x)
even function
left half reflects the line of the right halfgraph is symmetrical with respect to the y-axis; f(x) = f(-x)
Special Right Triangle 45-45-90
leg1 = leg2; hypotenuse = leg√2
point of inflection
point where concavity changes
Ordinal survey items
refers to oder (ranking in order)
derivative of tanx
sec^2(x)
inductive reasoning
specific to general
z-score
tells us how many standard deviations from the mean an observation falls, and in what direction (pos/neg)
null hypothesis
the hypothesis a researcher or experimenter will try to disprove or discredit.
Law of Large Numbers
the larger the number of individuals that are randomly drawn from a population, the more representative the resulting group will be of the entire population
incenter of a triangle
the point of concurrency of the angle bisectors of a triangle
orthocenter of a triangle
the point of concurrency of the three altitudes of a triangle
Product Rule
uv' + vu'
Mean of Discrete Distribution
weights each outcome xi according to its probability, pi
unit circle cos
x
Quadratic Formula
x = -b ± √(b² - 4ac)/2a
Circle Function
x^2 + y^2 = r^2
point slope form
y - y1 = m(x - x1)
unit circle tan
y / x
log function derivative
d/dx ln(u) = 1 / u
monotone non-linear function
data is always positive or negative
Derivative of e^u
e^u du
one-to-one function
every output has exactly one input and vice versa (horiz and vert line test!)
Chain Rule f(g(x))
f'(g(x)) x g'(x)
Trig Function Equation
f(x) = Atrig(Bx - C) + D
vertex form of a quadratic function
f(x) = a(x-h)^2 + k
standard form of a linear equation
Ax+By=C
Pascal's Triangle (used for)
Easy way to show combinations
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a unique product of prime numbers.
Combination Formula
ORDER DOES NOT MATTER (Read as n choose r)
Permutation Formula
ORDER MATTERS (Read it as n choose r)
"at Least one" rule
P(L) = 1 - P(not L)
alternative hypothesis
The hypothesis stating what the researcher is seeking evidence of. Directly contradicts the null hypothesis.
centroid of a triangle
The point of concurrency of the medians of a triangle (median is vertex to midpoint of opposite side)
Special Right Triangle 30-60-90
X:x/3:2x
unit circle sin
Y
removable discontinuity
a "hole" in a graph
What do each of the constants stand for? f(x) = Atrig(Bx - C) + D
a = amplitude, period = 2pi/b, c/b = phase shift, d = vertical shift
corrollary
a secondary inference or deduction from a main theorem
conjecture
a statement that has not been proved: an educated guess
correlation coefficient
a statistical measure of the extent to which two factors vary together, and thus of how well either factor predicts the other. Will b between -1 and 1. Closer to the extremes - stronger the correlation
Closed number system
a system is closed under an operation if performing that operation on two elements of the system results in another element of that system
axiom
a universal truth; an established rule (don't have to prove)
explanatory variable
a variable that we think explains or causes changes in the response variable (independent)
Additive Inverse
a+(-a)=0
multiplicative inverse of a complex number
a/(a^2 + b^2) - b(a^2 + b^2)i
Exponential Derivative Rule d/dx a^x
a^x * x' * ln(a)
determinant of a 2x2 matrix IAI
ad - bc
standard form of a quadratic equation
ax² + bx + c = 0
Nominal survey items
categorical survey items (think school, food)
derivative of sinx
cosx