Algebra 2 - Chapter 6 Flashcards

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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?

5,000 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?

2^24 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?

2^3 Signup and view all the answers

What is the annual interest formula?

y = ab^t Signup and view all the answers

What is the compound interest formula?

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

What does the Product Property of logarithms state?

log(b)3 + log(b)5 = log(b)15 Signup and view all the answers

What does the Quotient Property of logarithms state?

log(b)2 - log(b)1 = log(b)2 Signup and view all the answers

What does the Power Property of logarithms state?

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

What does the Exponential-Logarithmic Inverse Property state?

log(b)b^x = x Signup and view all the answers

What is the One to One Property of logs?

if log(b)x = log(b)y then x = y Signup and view all the answers

What is the One to One Property of Exponents?

b^x = b^y then x = y Signup and view all the answers

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

False Signup and view all the answers

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

2 Signup and view all the answers

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

16 Signup and view all the answers

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

1 Signup and view all the answers

What is the continue form of the compound interest formula?

A = Pe^{rt} Signup and view all the answers

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

approximately 8.66 years Signup and view all the answers

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

ln20 Signup and view all the answers

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

ln20/ln5 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).

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