Writing Exponential Functions

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 does the base 'b' in an exponential function represent?

Back

The base 'b' in an exponential function determines the rate of growth or decay. If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay.

Front

How do you identify an exponential function from a set of data points?

Back

To identify an exponential function from data points, look for a pattern where the ratio of consecutive outputs is constant, indicating a multiplicative relationship.

Front

What is the general form of an exponential growth function?

Back

The general form of an exponential growth function is f(x) = a * b^x, where 'a' > 0 and 'b' > 1.

Front

What is the general form of an exponential decay function?

Back

The general form of an exponential decay function is f(x) = a * b^x, where 'a' > 0 and 0 < 'b' < 1.

Front

How do you convert a word problem into an exponential function?

Back

Identify the initial value, determine the growth or decay factor, and express the relationship in the form f(t) = a * b^t, where 't' represents time.

Front

What is the significance of the y-intercept in an exponential function?

Back

The y-intercept of an exponential function occurs at (0, a), where 'a' is the initial value of the function when x = 0.

Front

How can you graph an exponential function?

Back

To graph an exponential function, plot the y-intercept, identify additional points by substituting x values, and draw a smooth curve that approaches the x-axis (for decay) or rises steeply (for growth).

Front

What is the effect of changing the base 'b' on the graph of an exponential function?

Back

Changing the base 'b' affects the steepness of the graph. A larger base results in a steeper curve for growth, while a smaller base (greater than 1) results in a flatter curve.

Front

What is the relationship between exponential functions and logarithms?

Back

Exponential functions and logarithms are inverse functions. If y = b^x, then x = log_b(y). This means that logarithms can be used to solve for the exponent in exponential equations.