Sketching Graphs of the Form <em>y = ^a/_x + q</em>

Sketch the graph of \(g(x) = \dfrac{2}{x} + 2\). Mark the intercepts and the asymptotes.

Examine the standard form of the equation

We notice that \(a > 0\) therefore the graph of \(g(x)\) lies in the first and third quadrant.

Calculate the intercepts

For the \(y\)-intercept, let \(x = 0\):

\begin{align*} g(x)=& \cfrac{2}{x}+2 \\ g(0)=& \cfrac{2}{0}+2 \end{align*}

This is undefined, therefore there is no \(y\)-intercept.

For the \(x\)-intercept, let \(y = 0\):

\begin{align*} g(x)& = \cfrac{2}{x}+2 \\ 0& = \cfrac{2}{x}+2 \\ \cfrac{2}{x}& = -2 \\ \therefore x& = -1 \end{align*}

This gives the point \((-1;0)\).

Determine the asymptotes

The horizontal asymptote is the line \(y = 2\). The vertical asymptote is the line \(x = 0\).

Sketch the graph

Domain: \(\{x:x\in \mathbb{R}, x\ne 0\}\)

Range: \(\{y:y\in \mathbb{R}, y\ne 2\}\)

Sketch the graph of \(y = \dfrac{-4}{x} + 7\).

Examine the standard form of the equation

We see that \(a < 0\) therefore the graph lies in the second and fourth quadrants.

Calculate the intercepts

For the \(y\)-intercept, let \(x = 0\):

\begin{align*} y& = \cfrac{-4}{x}+7 \\ & = \cfrac{-4}{0}+7 \end{align*}

This is undefined, therefore there is no \(y\)-intercept.

For the \(x\)-intercept, let \(y = 0\):

\begin{align*} y& = \cfrac{-4}{x}+7 \\ 0& = \cfrac{-4}{x}+7 \\ \cfrac{-4}{x}& = -7 \\ \therefore x& = \cfrac{4}{7} \end{align*}

This gives the point \((\dfrac{4}{7};0)\).

Determine the asymptotes

The horizontal asymptote is the line \(y = 7\). The vertical asymptote is the line \(x = 0\).

Sketch the graph

Domain: \(\{x:x\in \mathbb{R}, x\ne 0\}\)

Range: \(\{y:y\in \mathbb{R}, y\ne 7\}\)

Axis of symmetry: \(y=x+7\) and \(y=-x+7\)