# Ultimate AP Pre Calc Notes (copy)

Ch. 1 - The Basics

• Point-Slope Form:

• (y - y1) = m(x - x1)

• x = (-b ± √(b2 - 4ac)) / (2a)

• Average Rate of Change:

• m = (f(b) - f(a)) / (b - a)

• Difference Quotient:

• (f(x + h) - f(x)) / h

• Vertical Line Test f(x) is not a function if the vertical line intercepts with more than one point

• Perpendicular lines have negative reciprocals

Transformations

Linear

y = a(x -h) + k

y = a(x - h)² + k

Cubic

y = a(x - h)³ + k

Square Root

y = a√(x - h) + k

y = a(x - h)1/2 + k

Reciprocal

y = a(1 / (x - h)) + k

Exponential

y= ab(x - h) + k

Combinations of Functions

Sum

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

Difference

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

Product

(fg)(x) = f(x) * g(x)

Quotient

(f / g)(x) = f(x) / g(x), g(x) ≠ 0

Composite

f(g(x))

**The domain of a composite function is restricted by the domain of the input function

Inverse Functions

• The composition of inverse functions is equal to 0 f(f-1(x))=x

• To find inverse functions, replace f(x), or y, with the x-variable and x with a y- . Then, solve for y, or f-1(x).

Scatterplots

• The sum of the square of differences between the actual values and the model values is called the sum of the squared differences.

• The model that has the least sum is called the least squared regression line for the data.

• Residual: (e) the difference between the actual value and the value predicted by the model

• e = y - ŷ (Actual - Predicted)

• Correlation coefficient: (r) how close the best fit is to the the points

• Calculator: Stat Edit Stat 2nd Menu Linreg

Finding Types of Functions (Linear, Quadratic, Exponential, Neither)

• Using the given table, find AROC, the 2nd Difference, and Ratio

• All AROC are Equal → Linear

• All 2nd Difference are Equal → Quadratic

• All Ratios are Equal → Exponential

• Concave up AROC over equal length input value intervals is increasing

• Concave down AROC over equal length input value intervals is decreasing

Ch 2 - Polynomials & Rational Functions

• Graph of polynomial functions are continuous (no breaks, holes, or gaps)

• Extrema: the minimums and maximums of a function

• Relative/Local Extrema

• Absolute/Global Extrema

• ∅ = Empty set

• Between two real zeros, there must be at least one local min or max

• Even-degree polynomials have either a global min or global max

• Point of Inflection: occurs where the function changes from concave up to concave down, or vice versa

• ROC changes from increasing to decreasing

Limits

• End Behavior: what happens to the values of f(x) as x increases or decreases without bounds

• To find, use leading coefficient test (e.g. LC is odd/even, negative/positive)

Zeros of a Polynomial

• x = a zero/solution of polynomial

• (x - a ) factor of polynomial

• (a, 0) x-intercept of polynomial

Imaginary Numbers

• i = √(-1)

• Complex Number: a + bi

• Complex Conjugate: a - bi

• To rationalize imaginary numbers (in denominators), multiply by the conjugate

• Division Algorithm Theorem: f(x) = d(x)*g(x) + r(x)

• where f(x) is = dividend, d(x) = divisor, g(x) = quotient, and r(x) = remainder

• Remainder Theorem: If polynomial f(x) is divided by x-h, then the remainder is r = f(k)

• Factor Theorem: A polynomial f(x) has a factor (x-h) if and only if f(k) = 0

• Rational Root Test: If a polynomial f(x) has integer coefficients, then every rational zero of f(x) has the form p / q

• where p = factor of constant term and q = factor of leading coefficient

Rational Functions

• f(x) = N(x) / D(x) = anxn / bmxm

• VA(s) are zeros of the denominator, HA is determined by the degrees of N(x) and D(x)

• n < m y = 0

• n = m y = LC/LC

• n > m no HA (slant asymptote, if n < m by 1)

• SA is found by dividing N(x) by D(x)

Graphing Rational Functions

1. Simplify f(x), if possible, and list all restrictions

2. Find and plot y-intercepts

3. Find and plot the zeros of the numerator

4. Find the VA (zeros of denominator)

5. Find and sketch other asymptotes with a dashed line

6. Plot min. 2 points between asymptotes and one point beyond each x-intercept and VA

7. Use smooth curves to complete graph

Ch. 3 - Exponentials and Logs

• Log Form: y = loga x

• a must be >1 and positive

• x must be positive

• Exponential Form: ay = x

Prop. of Logs

Product Property

• loga(bc) = loga b + loga c

• ln(bc) = ln b + ln c

Quotient Property

• loga(b/c) = loga b - loga c

• ln(b/c) = ln b - ln c

Power Property

• loga (b)c = (c)(loga b)

• ln (b)c = (c)(ln b)

• Change of Base Formula:

• loga x = ln x / ln a

• loga x = logb x / logb a

Logistic Model

Sequences

Prop. of Successive Terms

Formulas

Arithmetic

Common difference

• an = a0 +dn

• an = ak = d(n-k)

Geometric

Common ratio

• gn = g0(rn)

• gn = gk(rn-k)

Calculating Interest

• Normal Interest: A = P(1 + (r / n))nt

• Compound Interest: A = Pert

Ch. 4 - Trig Functions (radians)

• Counter-clockwise (CCW) angle

• Clockwise (CW) angle → -θ

• Coterminal Angles: angles which share the same terminal side in standard positon

• There’s an infinite number of coterminal angles

• Arc Length Formula: s = θr

• Unit Conversion: π rad = 180°

Trigonometric Functions

Inverse Trig Functions

sin θ = y / r

csc θ = r / y

arcsin (y / r) = θ

cos θ = x / r

sec θ = r / x

arccos (x / r) = θ

tan θ = y / x

cot θ = x / y

arctan (y / x) = θ

**Solve the right triangle find all angles and sides

**Remember to add 2π, n∈ℤ if there’s no restricted range

The Unit Circle

Trig Identities/Formulas

• All Students Take Calculus: Mnemonic to remember in which quadrants the trig functions are positive

Parent Graphs

• Sine Formula: y = a(sin(bx - c)) + d

• Cosine Formula: y = a(cos(bx - c)) + d

• Amplitude: (a) half of the distance between min/max values

• Period (T) = 2π / b

• Phase Shift = c / b

**MISTAKE: The midpoint of one tangent cycle is at the origin

Inverse Trig Function

Domain

Range

y = arcsin x

[-1, 1]

[-π/2, π/2] Q1 & Q4

y = arccos x

[-1, 1]

[0, π] Q1 & Q2

y = arctan x

(-∞, ∞)

(-π/2, π/2) Q1 & Q4

Law of Sines

Law of Cosines

Ch. 9 - Parametric & Polar Functions

Eliminating the Parametric

1. In one parametric equation, solve for t

2. Substitute the equation for t in the other equation

3. Simplify

Polar Coordinates

Coordinate Conversion

• sin θ = y / x

• cos θ = x / r

• tan θ = y / x

Polar-to-Rectangular

• x = r(cos θ)

• y = t(sin θ)

Rectangular-to-Polar

• tan θ = y / x

• x2 + y2 = r2

Testing for Symmetry in Polar Equations

• Over line θ = pi/2: replace (r, θ) by (r, θ-π) or (-r, -θ)

• The polar axis: replace (r, θ) by (r, -θ) or (-r, π-θ)

• The pole: replace (r, θ) by (r, π+θ) or (-r, θ)