For y=kenx+a, identify the horizontal asymptote y=a, intercept (0,k+a), and whether the curve lies above or below the asymptote. Here k,n,a are integers and kn 0; if k=0 or n=0, handle the resulting constant function directly. For y=kln(ax+b), require ax+b>0. The vertical asymptote is x=- ba, with the graph only on the allowed side. Here k,a,b are integers with ka 0. The range is R; the zero is at ax+b=1. A negative k reflects vertically; a negative a reverses the allowed side. For y=3e-x-2, the domain is R, range y>-2, and the curve decreases. Its intercepts are (0,1) and (ln 32,0). For y=-2ln(3-2x), the domain is x< 32. The curve increases and its intercepts are (1,0) and (0,-2ln3).

For \(y=k\mathrm{e}^{nx}+a\), identify the horizontal asymptote \(y=a\), intercept \((0,k+a)\), and whether the curve lies above or below the asymptote. Here \(k,n,a\) are integers and \(kn\ne0\); if \(k=0\) or \(n=0\), handle the resulting constant function directly.

For \(y=k\ln(ax+b)\), require \(ax+b>0\). The vertical asymptote is \(x=-\dfrac ba\), with the graph only on the allowed side. Here \(k,a,b\) are integers with \(ka\ne0\). The range is \(\mathbb R\); the zero is at \(ax+b=1\). A negative \(k\) reflects vertically; a negative \(a\) reverses the allowed side.

For \(y=3\mathrm{e}^{-x}-2\), the domain is \(\mathbb R\), range \(y>-2\), and the curve decreases. Its intercepts are \((0,1)\) and \((\ln\frac32,0)\).

For \(y=-2\ln(3-2x)\), the domain is \(x<\dfrac32\). The curve increases and its intercepts are \((1,0)\) and \((0,-2\ln3)\).

Transformed exponential and logarithmic graphs

Transformed exponential and logarithmic graphs