Graphing Exponentials

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 a positive real number, and 'x' is the exponent.

Front

What is the general shape of the graph of an exponential function?

Back

The graph of an exponential function increases rapidly for positive values of x and approaches zero for negative values of x, creating a J-shaped curve.

Front

What does the base 'b' in an exponential function f(x) = a * b^x represent?

Back

The base 'b' determines the rate of growth or decay of the function. If b > 1, the function grows; if 0 < b < 1, the function decays.

Front

How do you find the y-intercept of an exponential function?

Back

To find the y-intercept, evaluate the function at x = 0. For f(x) = a * b^x, the y-intercept is (0, a).

Front

What is the y-intercept of the function y = 3^x + 4?

Back

The y-intercept is (0, 5) because f(0) = 3^0 + 4 = 1 + 4 = 5.

Front

What is the domain of an exponential function?

Back

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

Front

What is the range of an exponential function of the form f(x) = a * b^x where a > 0?

Back

The range is (0, ∞) because the function approaches zero but never reaches it.

Front

What is the horizontal asymptote of the function y = 3^x + 4?

Back

The horizontal asymptote is y = 4, as the function approaches this value for very large negative x.

Front

How can you determine if a function is exponential from its equation?

Back

A function is exponential if it can be expressed in the form f(x) = a * b^x, where 'a' is a constant and 'b' is a positive real number.

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.