Algebra Class: Functions and Slope

Slope Formula

• Slope between two points is calculated using the formula ((y₂-y₁)/(x₂-x₁)).

Point-Slope Form

• The equation of a line can be expressed as (y-y₁=m(x-x₁)), where ((x₁, y₁)) is a specific point, and (m) represents the slope.

Absolute Value Function

• The equation (y=a|x-h|+k) represents transformation of the absolute value graph:
  • (a) is the stretch factor (larger values narrow the graph).
  • ((h,k)) denotes the vertex; (h) shifts horizontally, while (k) shifts vertically.

Function Definition

• A function is defined as having exactly one output (y-value) for every input (x-value), verifiable through the vertical line test.

Exponent Rules

• Rules include:
  • ((xⁿ)(x°)=xⁿ⁺°)
  • ((xⁿ)/(x°)=xⁿ⁻°)
  • ((xⁿ)°=xⁿ°)

Logarithmic Definition

• The expression (Log_n(A)=P) can be rewritten in exponential form as (n^P=A), pronounced as "log nap".

Logarithmic Rules

• Key rules for logarithms:
  • (log(a/b) = log(a) - log(b))
  • (log(a)(b) = log(a) + log(b))
  • (log(xⁿ) = nlog(x))
  • (log(1) = 0) and (ln(1) = 0)

Exponential Equation

• The equation (y=ab^x) describes an exponential function where:
  • (a) is the initial value and (b) is the growth rate.

Continuous Exponential Growth

• Expressed as (y=ae^{kt}), where (a) is the initial value, and (k) represents the growth rate, which can be positive or negative.

Arithmetic Sequence Formula

• The nth term of an arithmetic sequence is found using (A_n = a₁ + (n-1)d):
  • (a₁) is the initial value, (d) is the common difference, and (n) is the term number.

Geometric Sequence Formula

• The nth term of a geometric sequence is given by (A_n = a₁(r)^{n-1}):
  • (a₁) stands for the initial value, (r) is the common ratio, and (n) denotes the term number.

Arithmetic Series Sum

• The formula for the sum of an arithmetic series is (S_n = .5n(2a₁ + [n-1]d)):
  • (a₁) is the starting value, (d) is the common difference, with (n) as the number of terms.

Geometric Finite Series Sum

• The sum of a geometric finite series is calculated using (S_n = a₁(1 - r^n)/(1 - r)):
  • (r) is the common ratio and (n) is the number of terms.

Geometric Infinite Series Sum

• The formula for the sum of a geometric infinite series is (S_n = a₁/(1 - r)):
  • Can only be applied when (|r| < 1) for convergence, where (a₁) is the initial value.