Exponents and Exponentials

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 the base (a positive real number), and 'x' is the exponent.

Front

What is the difference between exponential growth and exponential decay?

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Exponential growth occurs when the base 'b' is greater than 1 (b > 1), leading to an increase in value as 'x' increases. Exponential decay occurs when the base 'b' is between 0 and 1 (0 < b < 1), leading to a decrease in value as 'x' increases.

Front

What is the y-intercept of an exponential function?

Back

The y-intercept of an exponential function f(x) = a(b^x) is the value of f(0), which equals 'a'.

Front

How do you simplify a negative exponent?

Back

A negative exponent indicates a reciprocal. For example, a^(-n) = 1/(a^n).

Front

What is the common ratio in an exponential function?

Back

The common ratio in an exponential function f(x) = a(b^x) is the base 'b', which represents the factor by which the function value changes for each unit increase in 'x'.

Front

What is the formula for exponential growth?

Back

The formula for exponential growth is f(x) = a(1 + r)^x, where 'a' is the initial amount, 'r' is the growth rate, and 'x' is time.

Front

What is the formula for exponential decay?

Back

The formula for exponential decay is f(x) = a(1 - r)^x, where 'a' is the initial amount, 'r' is the decay rate, and 'x' is time.

Front

How do you find the initial value of an exponential function?

Back

The initial value of an exponential function f(x) = a(b^x) is found by evaluating f(0), which equals 'a'.

Front

What does it mean for a function to be increasing?

Back

A function is increasing if, as 'x' increases, the value of f(x) also increases. In exponential functions, this occurs when the base 'b' is greater than 1.

Front

What does it mean for a function to be decreasing?

Back

A function is decreasing if, as 'x' increases, the value of f(x) decreases. In exponential functions, this occurs when the base 'b' is between 0 and 1.