Exponential Transformations & Characteristics

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

Front

What is the y-intercept of an exponential function f(x) = a * b^x?

Back

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

Front

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

Back

A horizontal translation shifts the graph of the function left or right. For example, f(x) = a * b^(x - h) translates the graph h units to the right.

Front

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

Back

A vertical translation shifts the graph of the function up or down. For example, f(x) = a * b^x + k translates the graph k units up.

Front

What does a reflection across the x-axis mean for an exponential function?

Back

A reflection across the x-axis inverts the function, changing its sign. For example, f(x) = -a * b^x reflects the graph of f(x) = a * b^x.

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 0 < b < 1.

Front

How do you identify the negative interval of an exponential function?

Back

The negative interval is the range of x-values for which the function outputs negative values. For most exponential functions, this occurs when x is less than a certain value.

Front

What is the starting term 'a' in the function f(x) = 300(1.16)^x?

Back

The starting term 'a' is 300, which represents the initial value of the function when x = 0.

Front

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

Back

Changing the base 'b' affects the steepness of the graph. A larger base results in a steeper curve, while a smaller base (greater than 0) results in a flatter curve.