Chapter 8 Math Test Flashcards

Coordinate Solutions in Inequalities

• Solutions are within the shaded area or on a solid line of the graph.
• Points outside the shaded area or on a dashed line are not solutions.

Exponents

• Defined as a base (e.g., 3) raised to an exponent (e.g., ³).
• The expression 3³ is referred to as the power.

Addition of Exponents

• Combine like terms by adding coefficients in expressions such as 5x² + 4x² = 9x².
• Exponents and variables remain unchanged during addition.

Multiplication of Exponents

• When multiplying terms like (6x³)(2x³)(X³), multiply coefficients and add exponents: 12x⁹.
• Different variables can still be multiplied together.

Multiplying with Parentheses

• In an expression like (3x³)², square both the coefficient and the base while retaining the exponent: results in 9x⁶.

Division of Exponents

• For division, like 16x⁴ ÷ 8x², divide coefficients and subtract exponents: results in 2x².

Division with Negative Exponents

• Example: For -16x⁴ ÷ 8x², convert the result into a negative exponent by placing the variable with a negative exponent in the denominator: -2x⁻² = -2/x².

Scientific Notation

• Represents numbers as a decimal ≥ 1 and < 10 multiplied by a power of ten (10^n).
• Negative exponents indicate small numbers (decimal shifts left), positive exponents indicate large numbers (decimal shifts right).

Exponential Growth

• Characterized by a curve that approaches but never goes below zero.

Linear Function Characteristics

• Linear functions involve a consistent increase or decrease in Y-values for each unit increase in X.
• Equation format is Y = mx + b, where M = slope and B = Y-intercept.

Exponential Function Characteristics

• Involves multiplying Y-values as X-values increase by one.
• Equation format is Y = Ab^x, where A is the initial value and B indicates the growth rate.

Domain and Range of Exponential Functions

• Domain is all real numbers since any X-value can be input.
• Range is restricted to Y > 0, meaning Y-values cannot be negative.

Exponential Growth Model

• Described by the equation Y = A(1 + R)^T, where:
  • A = initial amount
  • R = growth rate
  • T = time in years

Exponential Decay

• Similar to exponential growth but trends downward.
• Represented by Y = A(1 - R)^T, where R is the decay rate.

Decay Factor

• Calculated as (1 - r), where r is the decay rate.

Ponderosa Pine Forests Example

• The model Y = 41,000,000(1 - 0.005)^39 calculates the remaining acreage over time, accounting for annual decay.

Negative Exponents Handling

• A negative exponent indicates a reciprocal. For example, x⁻² = 1/x².
• In fractions, negative exponents in the denominator are transformed by reciprocal adjustment.

Identifying Domain and Range

• Domain represents the set of possible X-values.
• Range represents the set of possible Y-values.

Expanded Form Representation

• A number's expanded form illustrates the contribution of each digit based on its place value (e.g., 3,333,000 indicates three millions).

Operations with Scientific Notation

• When dividing, divide coefficients and subtract the exponents of the powers of 10; for instance, 3.2 x 10³ ÷ 2.0 x 10² results in 1.6 x 10¹.

Exponential Growth and Decay Models

• Growth model Y = C(1 + R)^T represents costs where C is the initial cost.
• Decay equations focus on decreasing values where A is still greater than 0, and b must be a fraction between 0 and 1.