ap pre calc

Row 0, 1, and 2 of Pascal's Triangle

1

1 1

1 2 1

reflection over the y-axis

switch x values -> f(-x)

reflection over the x-axis

switch y values -> -f(x)

sin(2a) is equivalent to

2sin(a)cos(a) -> double angle

cos(2a) is equivalent to (name all 3)

cos²a - sin²a

1 - 2sin²a

2cos²a - 1

sin(a+b)

sin(a)cos(b) + sin(b)cos(a)

sin(a-b)

sin(a)cos(b) - sin(b)cos(a)

cos(a+b)

cos(a)cos(b) - sin(a)sin(b)

cos(a-b)

cos(a)cos(b) + sin(a)sin(b)

range of sin⁻¹(x)

[-π/2, π/2]

range of cos⁻¹(x)

[0, π]

range of tan⁻¹(x)

[-π/2, π/2]

general solution of sin(x)

x = θ + 2πk, where k is an integer

general solution of cos(x)

x = θ + 2πk, where k is an integer

general solution of tan(x)

x = θ + πk, where k is an integer

VAs of csc(x)

x = 0 + πk, where k is an integer

VAs of sec(x)

x = π/2 + πk, where k is an integer

VAs of cot(x)

x = 0 + πk, where k is an integer

range of csc(x)

(-∞, -1] U [1, ∞) -> -1 and 1 are affected by VERTICAL transformations

range of sec(x)

(-∞, -1] U [1, ∞) -> -1 and 1 are affected by VERTICAL transformations

range of cot(x)

(-∞,∞)

VAs are on…

the BOTTOM of a polynomial

zeros are on…

the TOP of a polynomial

holes are on…

both the top AND bottom of a polynomial

the domain of a polynomial excludes…

the VAs and holes

if n > d…

there is no HA

is n = d…

the HA is the first term of the numerator/the first term of the denominator

if n < d…

the HA is y = 0

a slant asymptote is only present when…

the numerator is exactly one degree greater than the denominator

polar to rectangular coordinate conversion

(r cos(θ), r sin(θ))

rectangular to polar conversion (r coordinate)

√(x² + y²)

rectangular to polar conversion (θ coordinate)

tan(θ) = y/x

sentence stem for when a table shows a LINEAR function

The average rates of change of the output values over equal length input values is _, which is constant

sentence stem for when a table shows a QUADRATIC function

The change in the average rates of change/the second differences of the output values over equal length input values is _, which is constant.

sentence stem for when a table shows a CUBIC function

The output values increase/decrease on __ < x < __, increase/decrease on __ < x < __, and increase/decrease again on __ < x < __ on equal length input values.

sentence stem for when a table shows a EXPONENTIAL function

The input values are added by ___ while the output values are multiplied by a factor of ___.

sentence stem for when a table shows a LOGARITHMIC function

The input values are multiplied by a factor of ___ while the output values are added by ___.

showing that a function can/can't be an inverse function

Every output value of the graph/table (does not) corresponds with/is mapped to a unique input value. (if the answer is no, give a specific point)

stating the average rate of change in formal terms

On average, increases/decreases by (units per time) from t = __ to t = __ (time)

at what interval are average rates of change larger/smaller

The average rates of change will be greater/smaller from t = __ to t = __ because the graph is (logarithmic, exponential, quadratic, etc) and is concave down/up, meaning the average rates of change are increasing/decreasing

how long is the model appropriate (increases/decreases by a certain number)

The model is appropriate for the first __ (time) because from __ (time) to __ (time), there is less than a __ (units) decrease/increase in (units).

how long is the model appropriate (goes under/over a certain number)

This model is appropriate for the first __ (time) because after ___ (time), the number of ___ (units) falls below/goes above __.

what point should experts have more confidence in

t = __ is what they should have more confidence in. It falls between the given data set used to create the model. We do not know what will happen after t = last data point's x-value (units).

decreasing or increasing rate of change

The interval is concave , so the rate of change is .