Practice Problem Set: Curve Sketching

What Curve Sketching Is Really About

Curve sketching is not about guessing what a graph should look like. It is about collecting enough information about a function to determine how its graph behaves. In AP Calculus, derivatives give you most of the information you need.

A strong curve-sketching solution usually combines the function's domain and intercepts with information from the first and second derivatives. You are looking for where the function increases or decreases, where it has relative extrema, where it is concave up or down, and whether it has asymptotes.

Think of the process as building the graph from evidence rather than trying to draw it from memory.

A Reliable Curve-Sketching Checklist

Find the domain. Identify values of that are excluded.
Find intercepts. Solve for the x-intercepts and evaluate for the y-intercept when appropriate.
Look for asymptotes. Check vertical, horizontal, and possibly oblique behavior.
Find the first derivative. Determine where is positive, negative, or zero/undefined.
Find critical numbers. These occur where or where does not exist, provided the function itself is defined there.
Determine increasing and decreasing intervals. Use a sign chart for .
Identify relative extrema. A sign change from positive to negative indicates a local maximum; a sign change from negative to positive indicates a local minimum.
Find the second derivative. Use to analyze concavity and possible inflection points.
Determine end behavior. Examine what happens as and near any vertical asymptotes.
Sketch the graph. Combine all of the information into one coherent picture.

The graph should come last

Do not begin by drawing a curve and then trying to justify it. First determine the mathematical facts. The sketch should be the final summary of those facts.

Worked Example 1: Polynomial Curve Sketch

Example 1: First and Second Derivatives

Sketch the graph of .

Step 1 — Domain: Since this is a polynomial, the domain is all real numbers: .

Step 2 — Intercepts: Factor the function: . Therefore, the x-intercepts are . The y-intercept is .

Step 3 — First derivative: .

Step 4 — Critical numbers: Set the derivative equal to zero: . Thus the critical numbers are .

Step 5 — Increasing/decreasing: The sign of gives: the function is increasing on and decreasing on .

Step 6 — Relative extrema: At , the derivative changes from positive to negative, so there is a relative maximum. At , it changes from negative to positive, so there is a relative minimum.

Step 7 — Concavity: . Therefore the graph is concave down when and concave up when .

Step 8 — Inflection point: The concavity changes at , giving the inflection point .

Conclusion: The intercepts, extrema, concavity, and end behavior together determine the overall S-shaped cubic graph.

Worked Example 2: A Rational Function

Example 2: Asymptotes and Derivative Analysis

Sketch the graph of .

Step 1 — Domain: The denominator cannot equal zero: . Thus is excluded from the domain.

Step 2 — Vertical asymptote: Since the denominator approaches zero while the numerator remains nonzero, the graph has a vertical asymptote at .

Step 3 — Horizontal asymptote: The degrees of the numerator and denominator are equal, so compare leading coefficients: . Therefore the horizontal asymptote is .

Step 4 — Intercepts: The x-intercept occurs when the numerator is zero: . The y-intercept is .

Step 5 — First derivative: Using the quotient rule: .

Step 6 — Increasing/decreasing: Because the denominator is squared, the sign of the derivative is determined by the numerator. Thus the function is increasing on and decreasing on .

Conclusion: The vertical asymptote divides the graph into separate branches, while the horizontal asymptote describes the behavior as . The derivative tells us how each branch moves between these features.

Worked Example 3: Exponential Curve Sketching

Example 3: Concavity and Horizontal Asymptote

Sketch the graph of .

Step 1 — Domain: The exponential function is defined for every real number: .

Step 2 — Intercepts: Since is always positive, there is no x-intercept. At , the y-intercept is .

Step 3 — First derivative: . Since the derivative is always positive, the function is increasing everywhere.

Step 4 — Second derivative: . This is also always positive, so the graph is concave up everywhere.

Step 5 — End behavior: As , the function approaches zero. Therefore is a horizontal asymptote.

Conclusion: The graph rises from left to right, remains concave up, passes through its y-intercept, and approaches its horizontal asymptote on the left.

How to Build a Sign Chart

Sign charts are one of the most useful tools for curve sketching. Instead of calculating dozens of function values, identify the important x-values and determine whether the derivative is positive or negative in each interval.

For the second derivative, the same idea determines concavity:

If , the graph is concave up.
If , the graph is concave down.
A candidate inflection point occurs where changes sign.

Common Mistakes

Finding critical numbers without checking the domain

A number where does not exist is a critical number only if the original function is defined there. A vertical asymptote is not a critical point.

Assuming every zero of the second derivative is an inflection point

A zero of is only a candidate. You must verify that the concavity actually changes from up to down or from down to up.

Using only the first derivative

The first derivative tells you about increasing/decreasing behavior and relative extrema. It does not by itself tell you the complete shape of the graph. Use the second derivative when concavity or inflection points matter.

Forgetting asymptotes

A graph can have the correct intercepts and derivative behavior but still be completely wrong if you ignore a vertical or horizontal asymptote.

Connecting points without respecting concavity

A curve should not simply be drawn through the important points. Its slope and concavity must agree with the derivative information you calculated.

Practice Problems

Question 1 — Polynomial

Analyze and sketch . Find the x-intercepts, critical numbers, intervals of increase/decrease, relative extrema, concavity, and inflection point.

Show answer

Domain: all real numbers.

First derivative: .

Critical numbers: .

Increasing: .
Decreasing: .

Second derivative: .

Concave down: .
Concave up: .

The concavity changes at , giving the inflection point .

Question 2 — Rational Function

Analyze and sketch . Identify the domain, intercepts, vertical and horizontal asymptotes, critical numbers, and intervals of increase/decrease.

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Domain: .

Vertical asymptote: .
Horizontal asymptote: .

x-intercept: .
y-intercept: .

Derivative: .
Since the derivative is positive wherever the function is defined, the function is increasing on both intervals of its domain.

Question 3 — Full Curve Analysis

For , determine the domain, intercepts, critical numbers, relative extrema, concavity, inflection points, and end behavior. Then sketch the graph.

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Domain: all real numbers.

First derivative: .
Therefore the critical number is .

The function is decreasing before the critical number and increasing afterward, so the critical point is a relative minimum.

Second derivative: . Thus the graph is concave up everywhere and has no inflection point.

As , the function approaches zero, giving a horizontal asymptote of .

Question 4 — Combined Analysis

Analyze the function . Construct a complete curve sketch using algebraic and derivative information.

Show answer

Domain: .

Intercepts: The x-intercepts occur at , and the y-intercept is .

First derivative: .
The critical numbers are .

Use the sign of the derivative to determine the increasing and decreasing intervals. The derivative changes from positive to negative at the first critical number, producing a relative maximum, and from negative to positive at the second critical number, producing a relative minimum.

Second derivative: . The concavity changes where , producing the corresponding inflection point.

A Fast AP Exam Strategy

On a free-response question, you usually do not need to calculate a large table of points. Instead, organize your work around the mathematical features that control the graph.

Start with the function itself: domain, intercepts, and asymptotes.
Use : critical numbers and increasing/decreasing behavior.
Use : concavity and possible inflection points.
Check end behavior: determine what happens far to the left and right.
Sketch last: make the curve satisfy every piece of information you found.

The most important habit is to treat the graph as a consequence of your calculations. If your derivative says the function is increasing, your sketch must rise. If the second derivative says the function is concave down, your sketch must bend accordingly. If there is an asymptote, the graph must respect it.

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