Exponential Functions - Graphs and Models

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 general form of an exponential function?

Back

The general form of an exponential function is f(x) = a * b^(x - h) + k, where (h, k) represents the horizontal and vertical shifts, respectively.

Front

What does the y-intercept of an exponential function represent?

Back

The y-intercept of an exponential function is the value of the function when x = 0, which is equal to a in the function f(x) = a * b^x.

Front

How do you determine the y-intercept of the function f(x) = 2 * 3^x?

Back

To find the y-intercept, substitute x = 0 into the function: f(0) = 2 * 3^0 = 2 * 1 = 2.

Front

What is a vertical translation in the context of exponential functions?

Back

A vertical translation involves shifting the graph of the function up or down by adding or subtracting a constant from the function.

Front

What does a vertical translation up by 2 units mean for the function f(x)?

Back

It means the new function can be expressed as f(x) + 2, shifting the entire graph of f(x) upward by 2 units.

Front

What is a horizontal translation in the context of exponential functions?

Back

A horizontal translation involves shifting the graph of the function left or right by adding or subtracting a constant from the x-variable.

Front

What does a horizontal translation left by 1 unit mean for the function f(x)?

Back

It means the new function can be expressed as f(x + 1), shifting the entire graph of f(x) to the left by 1 unit.

Front

What is the difference between exponential growth and exponential decay?

Back

Exponential growth occurs when the base b > 1, leading to an increase in the function's value as x increases. Exponential decay occurs when 0 < b < 1, leading to a decrease in the function's value as x increases.

Front

How can you identify exponential growth from a graph?

Back

A graph represents exponential growth if it rises sharply as x increases, showing a steep upward curve.