Summary of Exponential Function Transformations: Focus on Constant a

The standard form of the exponential function is:

f(x) = a · b^{(x - h)} + k

Core Focus: The Role of Constant a

a (vertical stretch/compression factor): This is the key focus of this knowledge point! a controls the vertical amplitude of the graph.

• If |a| > 1, the graph is vertically stretched (becomes "steeper" or "flatter" depending on growth/decay).
• If 0 < |a| < 1, vertically compressed (the graph is "squeezed").
• a > 0: Normal orientation; a < 0: Reflection over the x-axis (flipped).
• Key: a does not affect horizontal position (that's h), growth/decay rate (that's b), or vertical position (that's k). It only "scales" the output values of the exponential part.
• In real-world models, a often represents the initial quantity (e.g., initial temperature difference or population size); in assessments, it's commonly solved for using given points.

Common Errors (Specific to a)

• Mistaking a for affecting the growth rate (that's b).
• Treating a as k, leading to ignoring reflection or scaling effects.
• Forgetting to isolate a when solving (e.g., not subtracting k first).

Applications

In cooling models, a is the initial temperature difference; in population models, a is the starting population size. Practice by isolating a using given points.