APPC Flashcards

The “A” in g(x)=af(b(x+h))+k

Vertical Dilation by a factor of +a

Reflection over x-axis if a<0

The “C” in g(x)=af(b(x+c))+d

Horizontal Translation

Left when x+c

Right when x-c

The “d” in g(x)=af(b(x+c))+d

Vertical Translation up when +d, down when -d.

The “B” in g(x)=af(b(x+c))+d

Horizontal Dilation by a factor of +1/b

Reflection over the y-axis if b<0.

Average Rate of Change between (a,f(a)) and (b,f(b))

f(b)-f(a)/b-a

Where is a function positive/negative

Positive when y-coords above x-axis

Negative when y-coords below x-axis

What defines an increasing/decreasing function

Increasing when outputs increase as inputs increase

Decreasing when outputs decrease as inputs increase

What is a point of inflection

The ordered pair where concavity changes

What justifies an increasing rate of change

When a function is concave up

what justifies a decreasing rate of change

when a function is concave down

what does it mean if c is odd in f(x)=a(x-b)^c

c is a zero with odd multiplicity

graph of f will cross the x-axis at x=b

what does it mean if c is even in f(x)=a(x-b)^c

c is a zero with even multiplicity

the graph of f will touch the x-axis and turn at x=b

what is an even function

when f(x)=f(-x)

appears symmetric about the y-axis

what is an odd function

when f(-x)=-f(x)

appears symmetric about the origin

Notation for end behavior as inputs decrease without bound

lim f(x)

x→−∞

Notation for end behavior as inputs increase without bound

lim f(x)

x→infinity sign

Horizontal asymptote test when

r(x)=nth degree poly/nth degree poly

If m<n, HA @ y=0

If m=n, HA @ LCn degree/LCm degree

If n>m, no HA

When does a rational function have a slant asymptote

When degree of poly in numerator is exactly one more than the degree of the poly in the denominator

use long division to find

when does a rational function have a hole

when A factor cancels in numerator and denominator (unless covered by a VA)

example @ x=a in r(x)=(x-a)/(x-a)(x-b)

when does a rational function have a vertical asymptote

when A factor is a zero in the denominator after cancelling

example @ x=b in r(x)=(x-a)/(x-a)(x-b)

standard form of an arithmetic sequence

asubn=asubk+d(n-k) where (k,asubk) is any ordered pair in the sequence

standard form of a geometric sequence

gsubn=gsub1*r^(n-k) where (g,gsubk) is any ordered pair in the sequence

exponential decay in y=a*b^x

when 0<absolute value of b<1

as inputs increase, the outputs are moving towards the x-axis

exponential growth in y=a*b^x

when absolute value of b>1

as inputs increase, the outputs are moving away from the x-axis

logbasea of x+ logbasea of y

logbasea(xy)

nlogbasea of x=?

logbasea(x^n)

logbasea of x-logbasea of y

logbasea of (x/y)

logbasea of x/logbasea of y

logbasey of x

pythagorean trig identity

sin²x+cos²x=1

sin(alpha+beta)

sin alpha cos beta + cos alpha sin beta

sin(alpha-beta)

sin alpha cos beta - cos alpha sin beta

cos(alpha+beta)

cos alpha cos beta - sin alpha sin beta

cos(alpha-beta)

cos alpha cos beta + sin alpha sin beta

sin(2theta)

2 sin theta cos theta

cos(2theta)

cos^2 theta - sin^2 theta

2cos²theta-1

1-2sin²(theta)

Vertical asymptotes of f(x)=a*sec(bx)+d

bx=pi/2+pi(k), k is an element of the integers

Vertical asymptotes of f(x)=a*csc(bx)+d

bx=pi(k), k is an element of the integers

Vertical asymptotes of f(x)=a*cot(bx)+d

`bx=pi(k), k is an element of the integers

Vertical asymptotes of f(x)=a*tan(bx)+d

bx=pi/2+pi(k), k is an element of the integers

period of y=a*sin(b(x+c)) +d or y=a*cos(b(x+c)) +d

2pi/b

period of y=a*tan(b(x+c)) +d

pi/b

how to convert (r,theta)→(x,y)

x=rcostheta

y=rsintheta

hwo to convert (x,y)→(r,theta)

x²+y²=r², and:

tan theta=y/x, if x>0

tantheta=y/x+pi if x<0

Range of y=sin^-1x

bracket -pi/2, pi/2 bracket

how to convert a+bi to rcostheta +i(rsintheta)

a²+b²=r² and:

tantheta=b/a if a>0

tantheta=b/a+pi if a<0

Arc Length

theta*r

range of y=cos^-1x

bracket 0,pi bracket

range of y=tan^-1x

(-pi/2,pi/2)

In a polar function, when is distance from origin increasing

When:

r=f(theta) is positive and increasing

r=f(theta) is negative and decreasing

Concave up

In a polar function, when is distance from origin decreasing

When:

r=f(theta) is positive and decreasing

r=f(theta) is negative and increasing

concave down

what is error

absolute value of predicted value (from regression)-actual value

OR

The opposite sign of the residual

Compare period vs Frequency in a trig function

period is length required for one full cycle of outputs

Frequency is reciprocal of period

Frequency is how many cycles per unit of time

What type of function has constant rate of change in first differences over equal length inputs

A linear function

What type of function has constant rate of change in first differences over equal-length inputs

A quadratic function

What type of function has constant 3rd differences over equal length inputs

A cubic function

What type of function has proportional outputs over equal length inputs

An exponential function

What type of function has proportional inputs over equal length outputs

A logarithmic function

Average rate of change within the domain of a concave up model

overestimate

Average rate of change within the domain of a concave down model

underestimate

What type of data appear linear on a semi-log plot

Exponential