🎧 New: AI-Generated Podcasts Turn your study notes into engaging audio conversations. Learn more

Algebra 2 - Chapter 6 Flashcards
22 Questions
100 Views

Algebra 2 - Chapter 6 Flashcards

Created by
@GladLepidolite6058

Podcast Beta

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the form of a linear function?

  • y = mx + b (correct)
  • y = x^2 + x + 2
  • y = ab^t
  • none of the above
  • What is the equation for a quadratic function?

    x^2 + x + 2

    What is the form of an exponential function?

    2 = 3^x

    What is the initial amount (a) in the compound interest formula for $5,000 at 8% interest over 7 years?

    <p>5,000</p> Signup and view all the answers

    How many fruit flies are present after one day if there are 2 fruit flies that double every hour?

    <p>2^24</p> Signup and view all the answers

    How many fruit flies are present after 1 hour if there are 2 fruit flies that double every 20 minutes?

    <p>2^3</p> Signup and view all the answers

    What is the annual interest formula?

    <p>y = ab^t</p> Signup and view all the answers

    What is the compound interest formula?

    <p>A(t) = p(1 + r/n)^(nt)</p> Signup and view all the answers

    What does the Product Property of logarithms state?

    <p>log(b)3 + log(b)5 = log(b)15</p> Signup and view all the answers

    What does the Quotient Property of logarithms state?

    <p>log(b)2 - log(b)1 = log(b)2</p> Signup and view all the answers

    What does the Power Property of logarithms state?

    <p>log(b)m^p = p log(b)m</p> Signup and view all the answers

    What does the Exponential-Logarithmic Inverse Property state?

    <p>log(b)b^x = x</p> Signup and view all the answers

    What is the One to One Property of logs?

    <p>if log(b)x = log(b)y then x = y</p> Signup and view all the answers

    What is the One to One Property of Exponents?

    <p>b^x = b^y then x = y</p> Signup and view all the answers

    Is there a solution for the equation $2^x = -6$?

    <p>False</p> Signup and view all the answers

    What is the value of $e^{ln2}$?

    <p>2</p> Signup and view all the answers

    What is the value of $e^{4ln2}$?

    <p>16</p> Signup and view all the answers

    What is the value of $ln(e)$?

    <p>1</p> Signup and view all the answers

    What is the continue form of the compound interest formula?

    <p>A = Pe^{rt}</p> Signup and view all the answers

    How long will it take for an investment of $2,000 at 8% to double in value?

    <p>approximately 8.66 years</p> Signup and view all the answers

    If $e^x = 20$, what is the value of x?

    <p>ln20</p> Signup and view all the answers

    If $5^x = 20$, what is the value of x?

    <p>ln20/ln5</p> Signup and view all the answers

    Study Notes

    Functions Overview

    • Linear Function: Expressed as y = mx + b, where m represents the slope and b is the y-intercept.
    • Quadratic Function: Represented by the equation x^2 + x + 2; involves squared terms.
    • Exponential Function: Formulated as 2 = 3^x, indicating rapid growth.

    Interest Calculations

    • Compound Interest Formula: A(t) = p(1 + r/n)^(nt), where p is the principal amount, r is the rate, n is the number of times interest is compounded per year, and t is the time in years.
    • Annual Interest Formula: Similar to compound interest, expressed as y = ab^t, where ‘a’ is the initial amount, ‘b’ is the growth factor, and ‘t’ is time.

    Examples of Growth

    • Interest Example: Investing $5,000 at 8% interest over 7 years can be modeled using y = ab^t: a = 5,000, b = 1.08, t = 7.
    • Fruit Fly Growth: Starting with 2 fruit flies that double every hour results in 2^24 flies after 24 hours using y = ab^t (a = 2, b = 2, t = 24).
    • Fruit Fly Growth (Shorter Interval): If 2 fruit flies double every 20 minutes, this leads to y = ab^t with a = 2, b = 2, t = 3 (for 1 hour).

    Logarithmic Properties

    • Product Property: log(b)3 + log(b)5 = log(b)15; combines logarithms for multiplication.
    • Quotient Property: log(b)2 - log(b)1 = log(b)2; simplifies division within logarithms.
    • Power Property: log(b)m^p = p * log(b)m; exponents pulled in front as a multiplier.

    Inverse Properties

    • Exponential-Logarithmic Inverse Property: log(b)b^x = x; connects logarithms and exponents.
    • One to One Property of Logs: If log(b)x = log(b)y, then x must equal y; indicates uniqueness.
    • One to One Property of Exponents: If b^x = b^y, then x must equal y; also denotes uniqueness.

    Solutions and Limitations

    • No Solution Case: For an equation like 2^x = -6, there is no solution as the base must be positive.
    • Calculations with e: e^ln2 = 2 and e^(4ln2) = 16 by using properties of logarithms and exponents.
    • Natural Logarithm: ln(e) = 1.

    Time to Double Investment

    • Investment Doubling Time: For $2,000 at an 8% interest rate, to determine how long it takes to double:
      • Set up the equation 4,000 = 2,000e^(0.08t) leading to t = ln2 / 0.08, resulting in approximately 8.66 years.

    General Exponential Equations

    • Exponential Equations: Equations such as e^x = 20 can be solved using natural logarithms, resulting in x = ln(20).
    • Base Change: For equations like 5^x = 20, apply natural logarithm to isolate x: ln(5^x) = ln(20) results in x = ln(20)/ln(5).

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    Test your understanding of key concepts from Algebra 2 Chapter 6 with these flashcards. Explore definitions and formulas related to linear, quadratic, and exponential functions. Perfect for reviewing and reinforcing your algebra skills.

    More Quizzes Like This

    Use Quizgecko on...
    Browser
    Browser