Finding the constants of a curve from given points

When the equation of a curve has unknown constants and the curve passes through given points, each point gives one equation. Solve the equations, write the curve in full, and only then sketch it.

Method

1. Substitute each given point into the equation of the curve.
2. Solve for the constants, for example by subtracting one equation from the other.
3. Check the values: a base must be positive and not 1, and a logarithm needs a positive argument at each point.
4. Write the curve in full and sketch it with its asymptote and intercepts.

Example

The curve \(y=a(2^{-x})+b\) passes through \((0,\,5)\) and \((1,\,3)\). Find \(a\) and \(b\), then sketch the curve.

\(\begin{aligned} (0,\,5):\quad a+b&=5 \\ (1,\,3):\quad \tfrac12a+b&=3 \\ \tfrac12a&=2\quad\Rightarrow\quad a=4,\ b=1 \end{aligned}\)

So \(y=4(2^{-x})+1\): asymptote \(y=1\), \(y\)-intercept \((0,\,5)\), and no \(x\)-intercept, since \(y>1\) for every \(x\).