Algebra Class: Functions and Slope

Study Notes

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.

Description

This quiz covers essential concepts in algebra including the slope formula, point-slope form, exponential rules, functions, and logarithms. Test your understanding of these mathematical principles critical for mastering algebra. Perfect for high school students preparing for exams.

Algebra Class: Functions and Slope

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Questions and Answers

What is the equation to find slope from 2 points?

What is the equation of a line in point slope form?

What does the equation y=a|x-h|+k represent?

A function has one output for every input.

What are the exponent rules for multiplication and division?

What does log n(A)=P represent?

What are the logarithmic rules for division and multiplication?

What is the exponential equation where a represents initial value?

What is the equation for continuous exponential growth?

What does An=a₁+(n-1)d represent in mathematics?

What does An=a₁(r)ⁿ⁻¹ represent?

What is the formula for the sum of an arithmetic series?

What is the formula for a geometric finite series?

What is the formula for a geometric infinite series?