Exponential function transformations

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 it mean for an exponential function to be increasing?

Back

An exponential function is increasing if its output values grow larger as the input values increase, typically when the base 'b' is greater than 1.

Front

What is the effect of a vertical stretch on an exponential function?

Back

A vertical stretch occurs when the function is multiplied by a factor greater than 1, making the graph taller without changing its horizontal position.

Front

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

Back

A horizontal shift moves the graph left or right. For example, f(x) = a * b^(x - h) shifts the graph 'h' units to the right.

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 zero but never reaches it.

Front

What is the common ratio in an exponential function?

Back

The common ratio is the factor by which the function's output changes for each unit increase in the input. For f(x) = a * b^x, the common ratio is 'b'.

Front

How does a vertical compression affect an exponential function?

Back

A vertical compression occurs when the function is multiplied by a factor between 0 and 1, making the graph shorter.

Front

What does the term 'asymptote' refer to in the context of exponential functions?

Back

An asymptote is a line that the graph approaches but never touches. For exponential functions, the horizontal asymptote is usually y = 0.

Front

What is the transformation of the function y = 4^x to y = 2(4)^(x-3) + 1?

Back

The transformation includes a vertical stretch by a factor of 2, a horizontal shift to the right by 3 units, and a vertical shift up by 1 unit.

Front

What is the base of the exponential function f(x) = 2(4)^x?

Back

The base of the exponential function f(x) = 2(4)^x is 4.