da Muhammad Zamin mancano 2 anni
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Discontinuous Function - a function that contains a hole or break in the graph.
Continuous Function - a function that does not contain any holes or breaks in its graph
Even Function - any function that is symmetric about the y-axis
Odd Function - Function that is the same when rotated 180 degrees around the origin
Interval of decrease - the intervals in the domain where the y-values are getting smaller.
Interval of increase - the intervals in the domain where the y values are getting larger.
The domain and range switch as well
switch values of x and y and then solve for x
c determines the vertical transformation of the function
d determines the horizontal transformation of the function
1/k determines the horizontal stretch or compression from the original function
a determines the vertical stretch/compression from the parent function
y = a|k(x-d)|+c
y = a(1/(k(x-d))+c
y = asin(k(x-d))+c
y = a2^(k(x-d)) +c
y = mx+b
y=a(k(x-d))^2+c
Division: use the long division method
Normally divide the Formula using old school division method
When Factoring
Fourth step: Continue process until fully factored
Third step: divide the initial function
Second step: turn the x = y/z into zx - y = 0
First step: Find factors of c for which f(x) = 0
Multiplication: Each term of each Function is multiplied while their degrees are added
Subtraction: the y value of the second equation is subtracted from the y value of the first equation
Addition: their y values add up
the instantaneous rate of change is the rate of change on an exact point on the graph
(f(x+a)-f(x))/((a+x)-x): a is a small increment from the value of x(usually +0.01)
Graphically the slope of the line tangent to the point is the instantaneous rate of change
Tangent line: a line that just touches the exact point and avoids the points on the two sides of the point
the average rate of change is the rate of change of the graph over an interval
Formula: (f(x_2)-f(x_1))/(x_2-x_1)
Graphically a line must be made between the two points; the slope of the line is the average rate of change
the graph has at most n -1 number of turning points
When graphed: 0 to n(degrees of polynomial) number of intercepts
ordered based on a descending order of powers
Contain only one variable
Do not have vertical of Horizontal asymptotes
degrees is the highest power value of the polynomial
If the polynomial has 2n degrees(n does not = 0), then the two ends of polynomial will open towards same place(top or bottom), if not then they open in different directions
the increasing intervals of original turn to decreasing intervals in reciprocals and vice versa
the reciprocals of the x intercepts of the original is the x intercept of the reciprocal
the max and min of original become min and max in reciprocal
the reciprocal will intersect y = 1 or -1 if the original had it in its range
if original is linear or quadratic; horizontal asymptote will be y = 0
solving for x on the bottom results in the vertical asymptote of the equation
Horizontal asymptote is found by dividing the coefficients of p(x) by the coefficient of q(x)
if the degree of p(x) is one higher than the degree of q(x), there is an oblique asymptote
the vertical asymptotes: f(x) = p(x)/0
hole occurs at the x value where f(x) = 0/0
Pythagorean identities
1 + cot^2 θ = csc^2 θ
tan^2 θ + 1 = sec^2 θ
sin^2 θ + cos^2 θ = 1
Double angle formula
tan2θ = 2tanθ / 1 − tan^2 θ
cos2θ = cos2θ − sin2θ
sin2θ = 2 sin θ cos θ
Compound Angle Formulas
tan (a - b) = (tan(a) - tan(b))/(1 + tan(a) * tan(b))
tan (a + b) = (tan(a) + tan(b))/(1 - tan(a) * tan(b))
cos(a − b) = cos(a) cos(b) + sin(a) sin (b)
cos(a + b) = cos(a) cos(b) − sin(a) sin(b)
sin(a − b) = sin(a) cos (b) − cos(a) sin(b)
sin(a+b)= sin(a)cos(b)+cos(a)sin(b)
Tangent Graph
cos(x)
sin(x)
(positive a )+ c = max (negative) + c = min
c = equation of axis
(max+min)/2 = eoa
eoa is midpoint of max and min
k = 2pi/period
Positive a = amplitude
amplitude is half the distance between max and min
(max-min)/2=a
only cos is positive in 4th quad, all are positive in 1st quad, only sin is positive in 2nd quad, only tan is positive in 3rd quad
the range is {y E r}
the domain of the log function is {x E r / x > 0}
log function, but decreasing
3^x=3^3 then x=3
log_a(xy)=log_a(x)+log_a(y)
log_a(x/y)=log_a(x)-log_a(y)
log_a(x^n)=nlog_a(x)
log_a(1)=0
log_a(1/a^m)=-m
log_a(a^-m) = -m
a ^ log_a(x) = x
log_a(a^x)=x
log_a(M) = log_a(N) then M = N
ab^x=log(a)+log(b^x)
