Questions and Answers
How can you tell if a coordinate is a solution in a system of inequalities?
What does 3 represent in the exponent 3³?
base
What is the result of $5x² + 3x³ + 4x² + 8x³$?
9x² + 11x³
What do you do with the exponents when multiplying terms like $(6x³)(2x³)(X³)$?
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What is the result of $(3x³)²$?
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What is the result of $16x^4 ÷ 8x²$?
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How do you simplify $-16x^4 ÷ 8x²$?
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A decimal number that is greater than or equal to one and less than ten is expressed in ______ notation.
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What is the result of $-2(-2)²$?
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What is anything (except 0) to the zero power?
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What is expanded form?
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What is the result of $3.2 x 10³ ÷ 2.0 x 10²$?
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What is $(6.0 x 10³) (3 x 10³)$?
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What characterizes exponential growth?
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What defines a linear function?
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What is an exponential function characterized by?
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What is the domain and range for an exponential function?
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What is the exponential growth model formula?
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What does the decay factor represent in exponential decay?
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What does Y represent in the formula Y = A(1 - R) to the T?
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What is the decay equation?
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What was the decrease in acres of Ponderosa Pine Forests in 2002?
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What do you do when you encounter a negative exponent?
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What does 'Domain' refer to in a function?
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What does 'Range' refer to in a function?
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Study Notes
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.
Studying That Suits You
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Description
Test your knowledge with these flashcards covering key concepts from Chapter 8 of your math curriculum. Learn about identifying solutions in a system of inequalities, understanding exponents, and applying rules for combining terms. Each card provides essential definitions and examples to enhance your learning.
