I cannot provide you with a PDF answer key for Unit 6: Exponents and Exponential Functions. Sharing copyrighted material, including answer keys, is illegal and unethical.
However, I can help you understand exponents and exponential functions so you can solve the problems yourself. I can also provide examples to help you check your work.
Here's a breakdown of the key concepts you'll likely find in Unit 6:
Unit 6: Exponents and Exponential Functions - Key Concepts
Exponents:
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Definition: An exponent (or power) indicates repeated multiplication of a base number. For example, in 2³, 2 is the base and 3 is the exponent, meaning 2 × 2 × 2 = 8.
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Rules of Exponents: Mastering these rules is crucial:
- Product of Powers: xᵃ * xᵇ = x⁽ᵃ⁺ᵇ⁾ (When multiplying terms with the same base, add the exponents)
- Quotient of Powers: xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾ (When dividing terms with the same base, subtract the exponents)
- Power of a Power: (xᵃ)ᵇ = x⁽ᵃ*ᵇ⁾ (When raising a power to a power, multiply the exponents)
- Power of a Product: (xy)ᵃ = xᵃyᵃ (Distribute the exponent to each factor)
- Power of a Quotient: (x/y)ᵃ = xᵃ/yᵃ (Distribute the exponent to the numerator and denominator)
- Zero Exponent: x⁰ = 1 (Any nonzero base raised to the power of zero equals 1)
- Negative Exponent: x⁻ⁿ = 1/xⁿ (A negative exponent indicates the reciprocal)
- Fractional Exponent: x^(m/n) = ⁿ√(xᵐ) (The numerator is the power, and the denominator is the root)
Exponential Functions:
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Definition: An exponential function is a function where the independent variable (usually 'x') appears in the exponent. A common form is f(x) = a * bˣ, where 'a' is the initial value, 'b' is the base (and must be positive and not equal to 1), and 'x' is the exponent.
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Growth and Decay:
- Exponential Growth: If b > 1, the function represents exponential growth. The value of f(x) increases as x increases.
- Exponential Decay: If 0 < b < 1, the function represents exponential decay. The value of f(x) decreases as x increases.
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Graphing Exponential Functions: Exponential functions have characteristic curves. Growth functions increase rapidly, while decay functions decrease rapidly, approaching but never reaching zero.
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Applications: Exponential functions model various real-world phenomena, including population growth, compound interest, radioactive decay, and the spread of diseases.
To get help with specific problems:
Please provide the actual problems from your Unit 6 assignment. I can then guide you through the steps needed to solve them, explaining the concepts involved. Showing your work will allow me to identify any mistakes and help you understand where you went wrong. Remember, learning the process is more important than just getting the answers.
