math exam 1 Flashcards
(31 cards)
rational function equation
f(x) = [n(x)]/[d(x)]
3 cases of HA
- n > d : slant
- n = d : leading coefficients ratio
- n < d : y=0!
multiplicity
behavior of VA–
even: same direction, odd: opposite direction
behavior @ x-intercepts–
even: bounce, odd: cross
to find a hole
- factor n & d, simplify
- crossed out terms: the hole
- plug in that x to find y, coordinate!
to solve rational inequalities
- rearrange equation so simplified & equals 0
- finds the VAs (d) & the x-intercepts (n)
- make a number line with above values
- test intervals –> +? - ?
- write solution in interval form
linear inequalities
if ONE variable: set builder notation
if TWO variables: graph solution
vertex equation (for quadratics)
x = (- b)/2a
to graph quadratic inequalities
- find the vertex
- factor: find holes, intercepts (have at least THREE points!)
- connect the dots & label function
- test & shade the solution: (0,0) is the easiest test
to solve polynomial inequalities
- find x-intercepts & multiplicity
- make # line & test intervals
- interpret solution in interval notation (don’t include numbers that make d=0!)
to determine domain of a function
- start w/ all real numbers
- if there’s a denominator, exclude values that make it 0
- if there’s a radical of even index, don’t include values that make it negative
symmetry
even: f(-x)=f(x) –> across y-axis
odd: f(-x)= - f(x) –> across origin
neither: if f(-x) equals neither of above
to find an inverse
- swap x & y
- solve for “y”
- make y “f ^-1 (x)”!
to restrict domain
- break function @ vertex
- find each equation’s inverse
- revise inverse equation so applies to only that side
inverse info.
- always reflected over y=x from og function
- will pass the horizontal line test
transformations equation
y = af (bx + h) + k
a: vertical, stretch if >1
b: horizontal, stretch if 1>x>0
h: right if negative
exponential function equation
f (x) = Ca^x
pts: (-1, 1/a), (0, 1), (1, a)
HA: y=0
D: R, R: (0, oo)
to solve for an exponential function
- find slope twice: if same, linear; if not, exponential?
- find ratio of y to next y: is same for ALL, is exponential
- plug in a to find C
logarithmic function
inverse of exponential function
x = a^y equals y = log (base a) (x)
logarithmic properties (other)
- log # (1) = 0
- if base & x are matching = 1
- exponent rules:
log ^aM = M
log(base a) a^r = r
logarithmic properties
- y = log x^b –> = blogx
- log (1/M) –> -logM
- log (MN) –> logM + logN
- log (M/N) –> log M - log N
- logM=logN (same base) –> M = N
log on calc
“math” –> “logBASE”
log word problem equations
compounded continuously:
A=Pe^rt
compounded intervals:
A=P(1+r/n)^nt
A: future amount
P: principal investment
r: annual investment
t: time (in years)
n: # of times compounded per year
reference angle
smallest angle made by terminal side
w/ a sector, find theta
theta = s/r (arc length/radius)
