Ap PreCal Test Prep

A function is increasing on an interval if...

as the input values increase, the output value increase.

A function is decreasing on an interval if...

as the input values increase, the output value decrease.

A function is concave up if......

The rates of change are increasing.

A function is concave down if......

The rates of change are decreasing.

Average rate of change of f on the interval [a,b]

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

The slope of a function at any given point gives...

the rate of change of the function at that input.

A positive rate of change indicates that the function output is....

increasing

A negative rate of change indicates that the function output is....

decreasing

Point of inflection

point on the graph where the concavity changes, indicating a maximum or minimum rate of change.

one-to-one function

A function where each input has a unique output

A relative minimum occurs when a function f...

changes from decreasing to increasing

A relative maximum occurs when a function f...

changes from increasing to decreasing

Absolute minimum

the least output of a function

Absolute maximum

the greatest output of a function

Multiplicity

the number of times a factor occurs in a polynomial function

A polynomial of degree n has...

exactly n complex zeroes (real or imaginary) and at most n-1 extrema

If x=a is a real zero of a polynomial with an odd multiplicity , then...

The graph of the polynomial passes through the x-axis at x=a (slide for greater than 3)

If x=a is a real zero of a polynomial with an even multiplicity , then...

The graph of the polynomial bounces and is tangent at the x-axis at x=a

Odd Funtion

f(-x) = -f(x) rotational symmetry around the origin.

even function

f(-x)=f(x) symmetry across the y-axis.

End behavior of a polynomial f with an even degree and a negative leading coefficient

End behavior of a polynomial f with an odd degree and a positive leading coefficient

End behavior of a polynomial f with an odd degree and a negative leading coefficient

End behavior of a polynomial f with an even degree and a positive leading coefficient

If a rational function, f, has a horizontal asymptote at y=b, then...

Then both end behaviors approach b without bounds.

A rational function has a zero at x=a if ...

x=a is a zero of the numerator but NOT the denominator

For rational functions, a slant asymptote occurs when...

the degree of the numerator is exactly one more than the degree of the denominator

A function f(x) = ab^x demonstrates exponential growth if...

b>1

A function f(x) = ab^x demonstrates exponential decay if...

0<b<1

Key features of log where the base is greater than 1

Domain: x>0, range: All Real Numbers, VA @ x=0, increasing and concave down

Key features of exponential where the base is greater than 1

Domain: All real numbers, range: y>0, HA @ y=0, increasing and concave up

e^a(ln)b

b^a

logb(1)

logb(b)

logb(mn)

logb(m/n)

Pythagorean Identities

Reciprocal Identities

cos(A+B)

cos(A-B)

sin(A+B)

sinAcosB+cosAsinB

sin(A-B)

sinAcosB-cosAsinB

cos(2x)

sin(2x)

2sinxcosx

Given (x, y) in Cartesian coordinates, determine polar coordinates, (r, theta)

Given (r, theta) in polar coordinates, determine Cartesian coordinates, (x, y)

x = rsin(theta), y = rsin(theta)

A polar function r = f(theta) is increasing if...

as theta increase, r increases

A polar function r = f(theta) is decreasing if...

as theta increase, r decreases

The distance between a point on a polar function r=f(theta) and the origin is increasing if...

r is positive and increasing or r is negative and decreasing

The distance between a point on a polar function r=f(theta) and the origin is decreasing if..

r is positive and decreasing or r is negative and increasing

A function is linear if over equal-length input intervals, output values...

change by a constant amount

A function is quadratic if over equal-length input intervals, output values...

change by a constant second difference

A function is exponential if as input values change ________________, output values change _________________.

additively; multiplicatively

A function is logarithmic if as input values change ________________, output values change _________________.

multiplicatively; additively

The average rates of change on a linear function are...

constant

The average rates of change of a quadratic function...

are CHANGING at a constant rate or follow a linear pattern.

tan(theta) gives the __________ of the terminal ray of theta.

slope

Domain and range of y = arcsinx

Domain and range of y = arccosx

Domain and range of y = arctanx

f(x)= tan x has vertical asymptotes at ....

f(x)= cot x has vertical asymptotes at ....

Determine the amplitude, period, midline, and phase shift of f(x)=asin(b(x-c))+d

Amplitude: |a|
Period: 2pib (flip answer)
midline: y=d
phase shift: c units to the right

y = tan(bx) has a period of

pi*b

f(x) + c

vertical translation of c units

f(x-c)

horizontal translation of c units

cf(x)

vertical dilation of c

f(cx)

Horizontal dilation by a factor of 1/c

-f(x)

reflection over the x axis

f(-x)

reflection over y-axis

Pascal's Triangle

What does the constant e represent?

the base rate of growth for all continually growing processes, e~2.718

A positive residual indicates that the predicted value is an _________________________.

underestimate

A negative residual indicates that the predicted value is an _________________________.

overestimate