Comparing Exponential Functions

Front

What transformations occur when the function f(x) = 2^x is changed to g(x) = -2^(x+3) - 6?

Back

The transformations are: reflection over the x-axis, a horizontal shift left by 3 units, and a vertical shift down by 6 units.

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, and x is the exponent.

Front

What does it mean for an exponential function to have an asymptote?

Back

An asymptote is a line that the graph of a function approaches but never touches. For exponential functions, the horizontal asymptote is typically the value that the function approaches as x approaches infinity or negative infinity.

Front

What is the x-intercept of an exponential function?

Back

An exponential function does not have an x-intercept because it never crosses the x-axis; it approaches the x-axis asymptotically.

Front

How do you determine the rate of change of an exponential function?

Back

The rate of change of an exponential function can be determined by examining the base of the function; a larger base indicates a greater rate of change.

Front

What is the effect of a negative sign in front of an exponential function?

Back

A negative sign in front of an exponential function reflects the graph over the x-axis.

Front

How does a horizontal shift affect the graph of an exponential function?

Back

A horizontal shift moves the graph left or right along the x-axis, depending on the sign of the value added or subtracted from x.

Front

What is the significance of the base in an exponential function?

Back

The base of an exponential function determines the growth or decay rate; if the base is greater than 1, the function represents exponential growth, while a base between 0 and 1 represents exponential decay.

Front

What does it mean for an exponential function to be increasing or decreasing?

Back

An exponential function is increasing if its base is greater than 1, meaning the function's value rises as x increases. It is decreasing if the base is between 0 and 1, meaning the function's value falls as x increases.

Front

What is the vertical shift in an exponential function?

Back

A vertical shift moves the graph up or down along the y-axis, determined by adding or subtracting a constant from the function.