WARM-UP: Exponential Functions

Front

What does the base of an exponential function represent?

Back

The base of an exponential function represents the growth or decay factor. For example, in the function f(x) = a(b)^x, 'b' indicates how much the function increases (if b > 1) or decreases (if 0 < b < 1) for each unit increase in x.

Front

Define exponential growth.

Back

Exponential growth occurs when a quantity increases by a consistent percentage over equal time intervals. It can be modeled by the function f(x) = a(1 + r)^x, where 'r' is the growth rate.

Front

Define exponential decay.

Back

Exponential decay occurs when a quantity decreases by a consistent percentage over equal time intervals. It can be modeled by the function f(x) = a(1 - r)^x, where 'r' is the decay rate.

Front

What is the general form of an exponential function?

Back

The general form of an exponential function is f(x) = a(b)^x, where 'a' is a constant, 'b' is the base (growth/decay factor), and 'x' is the exponent.

Front

What does the value 'a' represent in the exponential function f(x) = a(b)^x?

Back

The value 'a' represents the initial amount or the y-intercept of the function when x = 0.

Front

How do you determine the domain of an exponential function?

Back

The domain of an exponential function is all real numbers, expressed as (-∞, ∞). This means that x can take any value.

Front

How do you determine the range of an exponential function?

Back

The range of an exponential function f(x) = a(b)^x (where a > 0) is (0, ∞) if b > 1 (growth) and (0, a) if 0 < b < 1 (decay).

Front

What is the effect of a growth rate of 6% on an investment?

Back

A growth rate of 6% means that the investment increases by 6% of its current value each period. This can be modeled by the function f(x) = a(1.06)^x.

Front

Write the exponential function for an investment of $1500 growing at 3.5%.

Back

B(t) = 1500(1.035)^t.

Front

What is the difference between exponential growth and linear growth?

Back

Exponential growth increases at a rate proportional to its current value, leading to faster increases over time, while linear growth increases by a constant amount.