Exponential transformations
\(y=Ae^{kx}+c\qquad A,k\ne0\)
| Feature | Result |
|---|---|
| Horizontal asymptote | \(y=c\) |
| \(y\)-intercept | \((0,A+c)\) |
| \(x\)-intercept | Solve \(Ae^{kx}+c=0\), if a real solution exists. |
\(y=2e^x+1\): asymptote \(y=1\), \(y\)-intercept \((0,3)\), no \(x\)-intercept since \(y>1\).
\(y=4-e^x\): asymptote \(y=4\), intercepts \((0,3)\) and \((\ln 4,\,0)\).
Logarithmic transformations
\(y=A\ln(x-h)+c\qquad A\ne0,\ x>h\)
| Feature | Result |
|---|---|
| Vertical asymptote | \(x=h\) |
| \(x\)-intercept | \((h+e^{-c/A},0)\) |
| \(y\)-intercept | \((0,A\ln(-h)+c)\), only if \(h<0\) |
For \(y=\ln(x+2)\): asymptote \(x=-2\), intercepts \((-1,0)\) and \((0,\ln2)\).
Mark the asymptote, domain and intercepts before sketching.
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