Pre Calc Midterm

Algebra review

Ch. 1

• Point-Slope Form:

  • (y - y₁) = m(x - x₁)

• Quadratic Formula:

  • x = (-b ± √(b² - 4ac)) / (2a)

• 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

**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).

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) = a_nxⁿ / b_mx^m

• 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 = log_a x

  • a must be >1 and positive

  • x must be positive

• Exponential Form: a^y = x

• Change of Base Formula:

  • log_a x = ln x / ln a

  • log_a x = log_b x / log_b a

Logistic Model

Calculating Interest

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

• Compound Interest: A = Pe_rt

Ch. 4 - Trig Functions (radians)

• Counter-clockwise (CCW) angle → +θ

• 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°

**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

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

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)