Exponential Function Transformations

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.

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

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

The range of an exponential function f(x) = a * b^x is (0, ∞) if a > 0, and (-∞, 0) if a < 0.

The parent function of exponential functions is f(x) = b^x, where b > 0.

A vertical shift of k units in the function f(x) = a * b^x + k moves the graph up if k > 0 and down if k < 0.

A horizontal shift of h units in the function f(x) = a * b^(x - h) moves the graph to the right if h > 0 and to the left if h < 0.

Stretching an exponential function means increasing the value of 'a' (the coefficient), making the graph steeper, while compressing means decreasing 'a', making the graph flatter.

A negative coefficient in an exponential function f(x) = -a * b^x reflects the graph across the x-axis.

This function is translated left 1 unit and down 3 units from the parent function f(x) = 2^x.