Curve Sketching

Curve sketching is the process of drawing an accurate graph of a function from its formula, using calculus to find every feature that matters. Each tool from this chapter answers one question:

• answers where the graph rises and falls, and where its peaks and valleys are.
• answers which way the graph bends, and where the bending switches.
• Limits answer what happens at the edges: asymptotes and end behavior.

A graphing calculator gives you a picture. Curve sketching tells you why the picture looks the way it does, and guarantees you haven’t missed a feature off-screen.

What f′ and f″ Tell You About a Graph

The sign of gives the direction, from the first derivative test. The sign of gives the bending, from concavity. Combining the two, every smooth piece of a graph has one of four shapes:

Curve sketching finds where those signs change, so you know which shape to draw on each interval.

Curve Sketching Steps

1. Domain. Where is defined? Note any excluded points.
2. Intercepts. The y-intercept is . The x-intercepts solve (skip them if they’re hard to find).
3. Symmetry. If , the graph is symmetric about the y-axis (even). If , it’s symmetric about the origin (odd).
4. Asymptotes and end behavior. Find vertical asymptotes with infinite limits and horizontal asymptotes with limits at infinity.
5. First derivative. Find the critical points, make a sign chart for , and list the intervals of increase and decrease and the local maxima and minima.
6. Second derivative. Make a sign chart for , and list the intervals of concavity and the inflection points.
7. Sketch. Plot the intercepts, extrema and inflection points, draw the asymptotes, and connect the points with the shapes the sign charts call for.

Example 1: A Cubic

Sketch .

Domain and intercepts. All real numbers. , and gives x-intercepts and .

Symmetry and asymptotes. Neither even nor odd. A polynomial has no asymptotes; as , , and as , .

First derivative. , so the critical points are and .

So has a local maximum at and a local minimum at .

Second derivative. , which is negative for and positive for . So is concave down on , concave up on , and has an inflection point at .

Sketch. Rise to the peak at , fall through the inflection point to the valley at , then rise through .

Example 2: A Quartic With a Flat Inflection Point

Sketch .

Intercepts. , and gives x-intercepts and . As , .

First derivative. . Critical points: and . Because , the sign of is the sign of : negative on and , positive on . So decreases until and increases after. The only local extremum is a minimum at . At the tangent is horizontal, but doesn’t change sign, so there’s no extremum there.

Second derivative. : positive on , negative on , positive on . Inflection points at and .

Sketch. Come down from the upper left, concave up. At the origin the graph flattens out (horizontal tangent) and switches to concave down while still falling. At it switches back to concave up, bottoms out at , and rises through .

Example 3: A Rational Function With Asymptotes

Sketch .

Domain. .

Intercepts and symmetry. The only intercept is . Since , the graph is symmetric about the y-axis.

Asymptotes. The denominator is zero at and the numerator isn’t, so there are vertical asymptotes there. Near : and ; by symmetry, the reverse happens at . The degrees of the numerator and denominator are equal, so , a horizontal asymptote .

First derivative. By the quotient rule,

The denominator is positive, so for and for (away from ). is increasing on and , decreasing on and , with a local maximum at .

Second derivative. Differentiating again and simplifying,

The numerator is always positive, so the sign follows : concave up for , concave down for . The concavity changes only at , which aren’t in the domain, so there are no inflection points.

Example 4: An Exponential Function

Sketch .

Domain and intercepts. All real numbers; the only intercept is .

End behavior. As , (by L’Hôpital’s rule), so is a horizontal asymptote on the right. As , .

First derivative. . Since , increases for and decreases for . Local (and absolute) maximum: .

Second derivative. . Concave down for , concave up for , with an inflection point at .

Sketch. Rise steeply from the lower left through the origin, peak at , fall and switch to concave up at , then flatten out toward the x-axis without reaching it.

Graphs of f, f′ and f″

Curve sketching also runs in reverse: given the graph of a function, sketch its derivative graph, or given the graph of , describe . The connections are the same sign facts, read off a picture instead of a formula.

Each graph tells you about the one above it in the same way: the sign of a graph is the direction of the graph above it.

How to sketch the graph of f′ from the graph of f

1. Find the horizontal tangents (peaks, valleys, flat spots) on . Mark those x-values as zeros of .
2. Fill in the signs. Where rises, draw above the x-axis. Where it falls, below.
3. Match the steepness. Where is steepest, is farthest from the axis. Near a peak or valley, is close to zero.
4. Use the inflection points. Where changes concavity, its slope is at a maximum or minimum, so has a peak or valley there.

You can also estimate individual values: is the slope of the tangent line at , so drawing the tangent and reading rise over run gives the derivative from a graph at that point.

Reading f from the graph of f′

Going the other way, look at where the graph of crosses the x-axis:

• Crosses from above to below: has a local maximum there.
• Crosses from below to above: has a local minimum.
• Touches the axis without crossing: horizontal tangent, but no extremum.
• Peak or valley of : inflection point of .

The graph of doesn’t tell you how high is, only its shape. You need one known value, such as , to place the graph vertically.

Sketch a Curve With the Following Properties

A common exercise gives you a list of properties instead of a formula.

Example 5. Sketch a curve with the following properties: , , ; on and ; on ; on and on .

1. Plot the given points , and .
2. Read . Rising, then falling at : local maximum at . Falling, then rising at : local minimum at .
3. Read . Concave down to the left of 0 and concave up to the right: inflection point at .
4. Connect with an increasing-concave-down piece up to , a decreasing-concave-down piece to , a decreasing-concave-up piece to , and an increasing-concave-up piece after.

The function has exactly these properties, but any curve with the same shape is a correct answer.

Common Mistakes

• Calling every critical point an extremum. Check that changes sign. In Example 2, is a flat inflection point, not a minimum.
• Calling every zero of an inflection point. The sign of must change, and the point must be in the domain (Example 3 has none).
• Connecting across a vertical asymptote. Treat each piece of the domain separately.
• Plotting only critical points. Intercepts and inflection points fix the shape between them.
• Mixing up the graph of with the graph of . A peak on is a zero on , not a peak.

For more worked sketches, see Section 4.5 of the free textbook OpenStax Calculus Volume 1.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. For , find the intervals where is increasing.

Problem 2. For , find the local maximum and minimum points.

Problem 3. For , find the inflection point.

Problem 4. Find all asymptotes of .

Problem 5. The graph of crosses the x-axis from below to above at . What happens to at ?

Problem 6. The graph of has a local maximum at . What does that tell you about ?

Curve sketching brings together every tool in this chapter. Next, the Integrals chapter starts with antiderivatives: undoing the derivative.