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Ultimate AP Pre Calc Notes (original)

Ch. 1 - The Basics

  • Point-Slope Form:

    • (y - y1) = m(x - x1)

  • Quadratic Formula:

    • 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

Quadratic

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

Half-Angle Quadrants

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)

CJ

Ultimate AP Pre Calc Notes (original)

Ch. 1 - The Basics

  • Point-Slope Form:

    • (y - y1) = m(x - x1)

  • Quadratic Formula:

    • 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

Quadratic

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

Half-Angle Quadrants

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)