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 vertical translation of an exponential function mean?

Back

Vertical translation refers to shifting the graph of the function up or down along the y-axis. For example, y = b^x + k translates the graph of y = b^x vertically by 'k' units.

Front

How does the base of an exponential function affect its growth or decay?

Back

If the base 'b' is greater than 1, the function represents exponential growth. If 0 < b < 1, it represents exponential decay.

Front

What is the horizontal asymptote of an exponential function?

Back

The horizontal asymptote of an exponential function is the line that the graph approaches as x approaches infinity or negative infinity. For functions of the form f(x) = a * b^x + k, the horizontal asymptote is y = k.

Front

What is the significance of the y-intercept in an exponential function?

Back

The y-intercept of an exponential function is the point where the graph intersects the y-axis, which occurs at (0, a) for the function f(x) = a * b^x.

Front

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

Back

To find the x-intercept, set f(x) = 0 and solve for x. For example, in f(x) = a * b^x, the x-intercept occurs when a * b^x = 0, which typically does not occur for positive 'a' and 'b'.

Front

What is the effect of adding a constant to an exponential function?

Back

Adding a constant to an exponential function results in a vertical translation of the graph. For example, y = b^x + k shifts the graph up by 'k' units.

Front

What is the difference between exponential growth and decay?

Back

Exponential growth occurs when the function increases rapidly as x increases (base > 1), while exponential decay occurs when the function decreases rapidly as x increases (0 < base < 1).

Front

What does the notation A = P(1 + r)^t represent?

Back

This notation represents the formula for compound interest, where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, and t is the time in years.

Front

What is the domain and range of an exponential function?

Back

The domain of an exponential function is all real numbers (-∞, ∞), and the range is (0, ∞) for functions of the form f(x) = a * b^x where a > 0.