Gradient Graphs

The gradient graph of \(y=\mathrm{f}(x)\) is \(y=\mathrm{f}'(x)\). Its value at each \(x\) is the gradient of the original graph.

A vertical asymptote remains at the same \(x\)-value.

A horizontal asymptote \(y=a\) becomes \(y=0\).

An oblique asymptote with positive gradient \(m\) becomes \(y=m\).

An oblique asymptote with negative gradient \(m\) becomes \(y=m\).

A minimum point \((a,b)\) becomes the \(x\)-intercept \((a,0)\).

A maximum point \((a,b)\) becomes the \(x\)-intercept \((a,0)\).

An increasing point of inflexion with gradient \(m\) becomes a minimum \((a,m)\).

A decreasing point of inflexion with gradient \(m\) becomes a maximum \((a,m)\).

Sketching from gradient regions

Partition the graph into regions of positive and negative gradient. Draw \(y=\mathrm{f}'(x)\) above the \(x\)-axis where \(\mathrm{f}\) is increasing and below it where \(\mathrm{f}\) is decreasing.

• If \(\mathrm{f}(x)\) approaches a horizontal asymptote, then \(\mathrm{f}'(x)\to0\).
• If \(\mathrm{f}(x)\) approaches the oblique asymptote \(y=mx+c\), then \(\mathrm{f}'(x)\to m\).
• If the gradient of \(\mathrm{f}(x)\) tends to infinity near \(x=a\), then \(y=\mathrm{f}'(x)\) has a vertical asymptote at \(x=a\).