Graphing Exponential Functions and Domain and Range

Front

What is an asymptote in the context of exponential functions?

Back

An asymptote is a line that a graph approaches but never touches. For exponential functions, the horizontal asymptote is often a constant value that the function approaches as x approaches infinity.

Front

What is the domain of an exponential function?

Back

The domain of an exponential function is all real numbers, represented as (-∞, ∞).

Front

What is the range of the function f(x) = a(b^x) + k, where a, b, and k are constants?

Back

The range is (k, ∞) if a > 0 and (−∞, k) if a < 0.

Front

What does it mean for a function to exhibit exponential growth?

Back

Exponential growth occurs when a function increases rapidly as x increases, typically represented by f(x) = a(b^x) where b > 1.

Front

What does it mean for a function to exhibit exponential decay?

Back

Exponential decay occurs when a function decreases rapidly as x increases, typically represented by f(x) = a(b^x) where 0 < b < 1.

Front

How do you identify the horizontal asymptote of the function f(x) = 2(1.5)^x + 7?

Back

The horizontal asymptote is y = 7, which is the constant added to the exponential function.

Front

What is the general form of an exponential function?

Back

The general form is f(x) = a(b^x) + k, where a is a constant that affects the vertical stretch, b is the base, and k is a vertical shift.

Front

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

Back

Changing the base 'b' affects the rate of growth or decay; larger bases lead to faster growth, while bases between 0 and 1 lead to decay.

Front

What is the vertical shift in the function f(x) = 2(1.5)^x + 7?

Back

The vertical shift is 7, which moves the graph up by 7 units.

Front

What is the significance of the constant 'a' in the function f(x) = a(b^x)?

Back

The constant 'a' determines the vertical stretch or compression of the graph and its direction (upward if a > 0, downward if a < 0).