4.1 Exponential Basics

Front

What is an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a * b^x, where a is a constant, b is a positive real number, and x is the exponent.

Front

What does it mean for a function to exhibit exponential growth?

Back

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to an increase that accelerates over time.

Front

What is the general form of an exponential growth function?

Back

The general form of an exponential growth function is f(x) = a(1 + r)^x, where a is the initial amount, r is the growth rate, and x is the time period.

Front

What is the difference between exponential growth and exponential decay?

Back

Exponential growth increases over time, while exponential decay decreases over time, typically represented by a function of the form f(x) = a(1 - r)^x.

Front

How do you simplify the expression 2^3 * 2^4?

Back

You can simplify it using the property of exponents: 2^3 * 2^4 = 2^(3+4) = 2^7.

Front

What is the x-intercept of an exponential function?

Back

An exponential function does not have an x-intercept because it never touches the x-axis; it approaches zero but never reaches it.

Front

What is the formula for compound interest?

Back

The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

Front

What is the base of an exponential function?

Back

The base of an exponential function is the constant factor that is raised to the power of the variable, typically denoted as b in the function f(x) = a * b^x.

Front

How do you determine if a graph represents exponential growth or decay?

Back

A graph represents exponential growth if it rises steeply as x increases, and it represents exponential decay if it falls steeply as x increases.

Front

What is the significance of the constant 'e' in exponential functions?

Back

The constant 'e' (approximately 2.718) is the base of natural logarithms and is used in continuous growth or decay models, represented as f(x) = ae^(bx).