Exponential Functions Overview

Parent Function Definition:

• The parent function for an exponential function is represented as y = b^x, where b > 1. This function serves as the basic form from which all exponential functions are derived.

Key Points of Exponential Functions:

• Exponential functions of the form y = b^x exhibit some essential characteristics:

  • Intercept: The graph intersects the y-axis at (0, 1), indicating that when x = 0, y = 1.

  • Growth: Rapid growth for x > 0, showing a constant ratio of change in y for equal changes in x.

  • Point of Value: At (1, b), y equals the base, illustrating direct proportionality.

Graphing:

• Ordered pairs derived from the function are plotted on the Cartesian coordinate plane.

  • For example, with y = 2^x:

  • (0, 1): Base case

  • (1, 2): Value at one unit in the positive x-direction

  • (-1, 0.5): Shows decrease and the asymptotic nature as it approaches zero.

Inverse Functions:

• The inverse function of y = b^x, represented as x = b^y (or y = log_b(x)), reflects across y = x.

• The graph of the inverse resembles a logarithmic curve, important properties include:

  • Approaching negative infinity as y decreases

  • Crossing the y-axis at (1, 0) and increasing without bound, demonstrating the inverse relationship.

Activity Focus:

• Analyze how the inverse resembles the contrails of the exponential function, focusing on y-intercepts and behavior in relation to axes, along with identifying local maxima and minima.