Everything AP PreCalculus

If a function is concave up

roc is increasing.

If a function is concave down

roc is decreasing.

Aroc -

constant rate of change for an interval of outputs.

Instantaneous rate of change -

rate that the y-values would change if the x-values changed at the point.

Iroc ≈

y₂ - y₁ / x₂ - (x₂ - 0.001)

The aroc of a linear function’s outputs are constant

over equal-length input-value intervals.

The aroc of a quadratic function’s outputs are constant

over consecutive equal-length input-value intervals.

The aroc of a nth function’s outputs are constant

over equal-length input-value intervals of an nth difference.

Degree -

the largest exponent in an equation.

Point of inflection -

where a function changes concavity.

Relative extrema -

where the polynomial switches direction/ a maximum or minimum but not the exact one.

Absolute/ Global extrema -

the greatest or least point in a function.

Odd multiplicity

intersects the x-axis.

Even multiplicity

is tangent to (bounces off) the x-axis.

X-intercept is a complex root if

(x^{^}² + #) = 0.

If a + bi is a root

then a - bi is a root.

Limit notation -

lim f(x) = #

x —> #

Limit notation if x approaches from the left:

lim f(x) = #

x^- —> #

Limit notation if x approaches from the right:

lim f(x) = #

x^+ —> #

Higher degree in the denominator so

there is a horizontal asymptote at y=0.

Higher degree in the numerator (by one #) so

the limits of numerator’s degree (and the leading coefficients).

Degree of the numerator and the denominator are equal so

there is a slanted asymptote at y = LC/LC.

Zeroes in the numerator are

zeroes.

Zeroes in the denominator are

undefined.

For inequalities check if

each range between each x-intercept/factor is positive or negative and included or not.

Holes (points of removable continuity) are present if

a factor of denominator simplifies out of the denominator.

The remainder in polynomial division is

the slant asymptote.

|a| in a transformation function is

a vertical dilation by |a|.

a<0 in a transformation function is

a reflection over the x-axis.

|b| in a transformation function is

a horizontal dilation by 1/|b|.

b<0 in a transformation function is

a reflection over the y-axis.

h (or c) in a transformation function is

a horizontal translation by -h (or -c).

k (or d) in a transformation function is

a vertical translation by k (or d).

The aroc of a piecewise function’s outputs are constant

over different equal-length input-value intervals.

Arithmetic -

successive terms with a common difference.

Geometric -

successive terms with a common ratio.

The equation for an arithmetic sequence:

a_{(of n)} = a_{(of k)} + d(n - k)

The a _{(of n)} of a_{(of n)} = a_{(of k)} + d(n - k) is

the value of the nth term of the arithmetic.

The d of a_{(of n)} = a_{(of k)} + d(n - k) is

the common difference between the term values.

The n of a_{(of n)} = a_{(of k)} + d(n - k) is

the nth term.

The equation for a geometric sequence:

g_{(of n)} = g_{(of k)} * r^{^(n - k)}

The g_{(of n)} of g_{(of n)} = g_{(of k)} * r^{^(n - k)} is

the value of the nth term.

The n of g_{(of n)} = g_{(of k)} * r^{^(n - k)} is

the nth term.

The r of g_{(of n)} = g_{(of k)} * r^{^(n - k)} is

the common ratio between the values.

The aroc of an exponential function’s output values are constant

proportionally over equal-length input-value intervals.

The a of f(x) = a(b)^x is

the initial amount.

The b of f(x) = a(b)^x is

the common ratio.

If (a > 0 and) b > 1 in f(x) = a(b)^x then f(x) is

a growth function.

If (a > 0 and) 1 > b > 0 in f(x) = a(b)^x then f(x) is

a decay function.

Exponential functions have a

horizontal asymptote.

b^m * b^n =

b^(m + n)

b^m / b^n =

b^(m - n)

(b^m)^n =

b^(m * n)

b^-m =

1 / b^m

Regression -

line of best fit.

Residual -

actual output minus the predicted output.

An overestimated regression has

a predicted value too large; a residual below the regression.

An underestimated residual has

a predicted value too small; a residual above the regression.

The points of a good regression model are

scattered.

f of g of x:

f(g(x)) or (f o g)(x)

To find the inverse

swap x and y, and solve for y.

Inversed equations are reflected

over y = x.

A function is invertible if

f(g(x)) = g(f(x))

Common log -

log10 or log.

Natural log -

loge or ln.

The limits of b of logb(c) = a are

b > 0 and b =/ 1.

The limits of c of logb(c) = a are

c > 0.

The base of logb(c) = a is

b.

The argument of logb(c) = a is

a.

The exponentional function of logb(c) = a is

b^a = c.

The aroc of a logarithmic function’s output are constant

multiplicatively over equal-length input-value intervals.

Log product property:

logb(x * y) = logb(x) + logb(y)

Log quotient property:

logb(x / y) = logb(x) - logb(y)

Log power property:

logb(x ^ y) = y * logb(x)

Change of base:

logb(x) = loga(x) / loga(b)

When the y-axis of a semi-log plot is scaled logarithmically an exponential function appears

linear.

To solve for a and b, y = logn(b) * x + logn(a) can be written as

log(y) = log(a) + log(b) * x.

Periodic relationship -

when the output values of a function demonstrate a repeating pattern over a successive equal-length input-value interval.

Period -

the length of the x-values that it takes for the function to complete on cycle.

f(x + nk) is f(x) where k is the period and

n is any integer (n ∈ℤ).

Angle measure in radians =

arc length / radius

Standard position -

an angle with its vertex at the origin and its initial ray along the positive x-axis.

Terminal ray -

the second ray forming an angle in standard position.

sinθ =

opposite / hypothenuse and y / r

cosθ =

adjacent / hypothenuse and x / r

tanθ =

opposite / adjacent and y / x

The coordinates of a point on a circle are

(r * cosθ, r * sinθ)

The coordinates in quadrant I are

(+, +).

The coordinates in quadrant II are

(-, +).

The coordinates in quadrant III are

(-, -).

The coordinates in quadrant IV are

(+, -).

Sinusoidal function -

a function that involves additive and multiplicative transformations of sinθ.

Amplitude -

the distance from the midline to the maximum or minimum.

Frequency =

1 / period

Midline -

halfway between the maximum and minimum.

The concavity of a sinusoidal function

oscillates.

Sine and cosine functions are

horizontal translations of each other.

Parent cosθ is

an even function.

Parent sinθ is

an even function.

An odd function is

f(x) = -f(-x).