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Section 6.4 Curve sketching

We now have all the information needed to sketch the graph of complicated functions by hand. In this section we provide a ten-step method for sketching the graph of a function \(f\text{,}\) using the information provide by \(f\text{,}\) its derivative \(f'\text{,}\) and its second derivative \(f''\text{.}\)

Subsection 6.4.1 Instructional videos

Subsection 6.4.2 Key concepts

Concept 6.4.1. Increasing/Decreasing Test.

Given a function \(f\text{,}\) one can find the intervals where the function is increasing and decreasing using information from \(f'\text{:}\)

  • If \(f'(x) \gt 0\) on an interval, then \(f\) is increasing on that interval.
  • If \(f'(x) \lt 0\) on an interval, then \(f\) is decreasing on that interval.
Concept 6.4.2. First Derivative Test.

The first derivative test is useful to determine whether a critical number of a function \(f\) is a local min or max, using information from \(f'\text{.}\)

Let \(c\) be a critical number of a continuous function \(f\text{.}\)

  • If \(f'\) changes from positive to negative at \(c\text{,}\) then \(f\) has a local max at \(c\text{.}\)
  • If \(f'\) changes from negative to positive at \(c\text{,}\) then \(f\) has a local min at \(c\text{.}\)
  • If \(f'\) does not change sign at \(c\text{,}\) then \(f\) has no local extremum at \(c\text{.}\)
Concept 6.4.3. Concavity Test.

Given a function \(f\text{,}\) one can find the intervals where the function is concave up or down using information from \(f''\text{:}\)

  • If \(f''(x) \gt 0\) on an interval, then \(f\) is concave up on that interval (happy).
  • If \(f''(x) \lt 0\) on an interval, then \(f\) is concave down on that interval (unhappy).

A point where the graph of a function is continuous and where the concavity changes is called an inflection point of \(f\text{.}\)

Concept 6.4.4. Second Derivative Test.

The second derivative test is useful to determin whether a critical number of a function \(f\) is a local min or max, using information from \(f''\)

Suppose \(f''\) is continuous near \(c\text{:}\)

  • If \(f'(c) = 0\) and \(f''(c) \gt 0\text{,}\) then \(f\) has a local min at \(c\text{.}\)
  • If \(f'(c) = 0\) and \(f''(c) \lt 0\text{,}\) then \(f\) has a local max at \(c\text{.}\)

Note that if \(f'(c)=0\) and \(f''(c) = 0\text{,}\) then the test is not conclusive: \(f\) may have a local max, a local min, or neither.

Concept 6.4.5. Vertical asymptotes.

The vertical line \(x=a\) is a vertical asymptote of \(y=f(x)\) if either

\begin{equation*} \lim_{x \to a^+} f(x) = \pm \infty \qquad \text{or} \qquad \lim_{x \to a^-} f(x) = \pm \infty. \end{equation*}
Concept 6.4.6. Horizontal asymptotes.

The horizontal line \(y=L\) is a horizontal asymptote of \(y=f(x)\) if either

\begin{equation*} \lim_{x \to \infty} f(x) = L, \qquad \text{or} \qquad \lim_{x \to - \infty} f(x) = L. \end{equation*}
Concept 6.4.7. Slant asymptotes.

Slant asymptotes occur when a function \(y=f(x)\) approaches a line (that is not horizontal) as \(x\) goes to \(\pm \infty\text{.}\)

The line \(y=m x + b\text{,}\) \(m \neq 0\text{,}\) is a slant asymptote of \(y=f(x)\) if either

\begin{equation*} \lim_{x \to \infty} \left( f(x) - (m x + b) \right) = 0 \qquad \text{or} \qquad \lim_{x \to - \infty} \left( f(x) - (m x + b) \right) = 0. \end{equation*}

This says that the vertical distance between the graph of \(y=f(x)\) and the line \(y=mx + b\) approaches \(0\) as \(x \to \infty\) or \(x \to -\infty\text{.}\)

Slant asymptotes are commonly found when \(f(x)\) is a rational function, and the degree of the numerator is one more than the degree of the denominator. Then, it can be found by performing long division for the rational function.

Concept 6.4.8. Summary of curve sketching.

Ten step method:

Information from \(f(x)\):

1. Domain:
find the domain of \(f\text{.}\)
2. Intercepts:
Find the \(y\)-intercepts \((0,f(0))\) and \(x\)-intercepts (the points where \(f(x) = 0\)).
3. Positivity:
Find where \(f\) is positive and negative.
4. Symmetry:
Is \(f\) even or odd? Is \(f\) periodic?
5. Asymptotes:
Find the vertical, horizontal and slant asymptotes of \(f\) if any exist.

Information from \(f'(x)\text{:}\)

6. Intervals of increase and decrease:
Find \(f'(x)\) and determine when \(f'(x) \gt 0\) (\(f\) is increasing) and when \(f'(x) \lt 0\) (\(f\) is decreasing).
7. Local max and min points:
Find the critical points of \(f\text{,}\) if any, and identify the local max and min.

Information from \(f''(x)\text{:}\)

8. Concavity:
Find \(f''(x)\text{,}\) determine when \(f''(x) \gt 0\) (\(f\) is concave up) and when \(f''(x) \lt 0\) (\(f\) is concave down).
9. Inflection points:
Find the inflection points of \(f\text{,}\) if any.

And finally...

10. Sketch the graph:
Plot key points (intercepts, critical points, local max and min, points of inflection), and sketch the curve together with its asymptotes.

Further readings 6.4.3 Further readings