Constant Ratio (for exponential functions): For an exponential function \(f(x)=ab^x\), the base \(b\) is the constant ratio by which the output is multiplied when the input increases by 1.
Theorems
Properties of the Graphs of Exponential Functions \(f(x)=b^x\) (where \(b>0, b \neq 1\)):
One-to-one function.
Horizontal asymptote: \(y=0\).
Domain: \((-\infty, \infty)\).
Range: \((0, \infty)\).
\(x\)-intercept: none.
\(y\)-intercept: \((0,1)\).
Increasing if \(b>1\).
Decreasing if \(0 < b < 1\).
Common Mistakes
Incorrect Order of Transformations: Applying shifts, reflections, and stretches/compressions in the wrong order, which can lead to an incorrect final graph. For example, applying a vertical shift before a vertical stretch will result in the shift also being stretched.
Errors in Horizontal Shifts: Confusing the direction of horizontal shifts (e.g., \(f(x+c)\) shifts left, \(f(x-c)\) shifts right for \(c>0\)).
Misinterpreting Stretches vs. Shifts: Confusing the effect of multiplying the function/input (stretch/compression) with adding to the function/input (shift).
Incorrectly Identifying the Asymptote after Transformations: Forgetting that vertical shifts change the horizontal asymptote, while horizontal shifts, reflections, and vertical stretches/compressions (by a non-zero factor) do not change the location of the horizontal asymptote \(y=0\) unless there's also a vertical shift.
Sign Errors with Reflections: Incorrectly applying reflections; \(-f(x)\) reflects across the \(x\)-axis, and \(f(-x)\) reflects across the \(y\)-axis.
Errors in Graphing Key Points after Transformation: Incorrectly transforming key points like the \(y\)-intercept or the point \((1,b)\) from the parent function.