Curve sketching uses calculus to predict the shape of a graph before plotting many points. This cheat sheet helps students organize first derivative tests, second derivative tests, intercepts, and asymptotes into one clear process. It is useful for checking calculator graphs, solving optimization-style problems, and explaining why a function rises, falls, bends, or levels off.
The reference is designed for quick review while practicing AP or precalculus-to-calculus graph analysis.
The first derivative tells where a function is increasing or decreasing and helps locate local extrema. The second derivative tells where a function is concave up or concave down and helps identify possible inflection points. Asymptotes describe end behavior or undefined behavior, including vertical, horizontal, and slant asymptotes.
A complete sketch combines domain, intercepts, critical points, concavity, asymptotes, and a few test values.
Key Facts
- Critical numbers occur where or where is undefined, as long as is in the domain of .
- If on an interval, then is increasing on that interval.
- If on an interval, then is decreasing on that interval.
- A local maximum can occur where changes from positive to negative, and a local minimum can occur where changes from negative to positive.
- If on an interval, then is concave up on that interval.
- If on an interval, then is concave down on that interval.
- A possible inflection point occurs where or is undefined, but the concavity must change there.
- A horizontal asymptote can be found by evaluating and when those limits are finite.
Vocabulary
- Critical Number
- A value in the domain of where or does not exist.
- Increasing Interval
- An interval where the function values rise as increases, usually shown by .
- Local Extremum
- A local maximum or local minimum where a function is higher or lower than nearby function values.
- Concavity
- The bending direction of a graph, determined by whether or .
- Inflection Point
- A point on the graph where concavity changes from up to down or from down to up.
- Asymptote
- A line that the graph approaches, often found using limits or undefined values of the function.
Common Mistakes to Avoid
- Treating every solution of as a maximum or minimum is wrong because the derivative must change sign or another test must confirm the extremum.
- Forgetting points where is undefined is wrong because corners, cusps, or vertical tangents can also create critical numbers.
- Calling every solution of an inflection point is wrong because concavity must actually change across that value.
- Ignoring the domain when finding asymptotes is wrong because excluded -values and discontinuities control where vertical asymptotes may occur.
- Sketching before making a sign chart is wrong because the signs of and determine increasing intervals, decreasing intervals, and concavity.
Practice Questions
- 1 For , find the critical numbers and determine where is increasing or decreasing.
- 2 For , find the intervals of concavity and all inflection points.
- 3 For , identify the vertical asymptote and determine whether the function has a slant asymptote.
- 4 Explain why a point where is not automatically an inflection point.
Understanding Curve Sketching Reference
Start with the domain because every later conclusion depends on where the original function exists. A denominator of zero, an even root of a negative number, or a logarithm of a nonpositive number can create restrictions. Mark excluded x values before making a sign chart.
An excluded value may produce a vertical asymptote, but not always. If a factor cancels during simplification, the graph can have a hole instead.
The simplified expression helps reveal nearby behavior, while the original expression tells whether the point belongs to the graph. This distinction prevents a common error in rational functions.
A sign chart is more reliable than trying to read everything from a derivative formula at once. Place every important x value in order on a number line. These include critical numbers, domain breaks, and endpoints when the problem gives a closed interval.
Pick one test value from each interval and determine the sign of the first derivative there. The actual derivative value matters less than its sign. Then connect the information to the original function.
A point where the derivative is zero is not automatically a high point or low point. For example, a graph can flatten briefly while continuing upward. The sign change is the evidence that decides the local behavior.
The second derivative describes how the slope itself changes. Concave up means slopes are becoming more positive as x moves right. A graph can be decreasing while concave up if its negative slopes are getting closer to zero.
Concave down means slopes are becoming more negative. This is useful when a sketch seems uncertain between two possible shapes. An inflection point requires a real switch in bending direction, not merely a second derivative equal to zero.
The point must lie on the original graph as well. In motion problems, concavity connects to acceleration. A position graph that is concave up has positive acceleration, though the object may still be moving left at that moment.
End behavior needs its own check because a few nearby points cannot show what happens far from the origin. For rational functions, compare the highest power in the numerator with the highest power in the denominator. This often predicts whether the graph approaches zero, approaches a constant, or follows a slanted line.
A graph may cross a horizontal asymptote because an asymptote describes far away behavior rather than a forbidden height. Vertical asymptotes need one sided thinking. The graph can rise on one side of a restricted x value and fall on the other.
After collecting all evidence, sketch each interval separately. Do not force a smooth connection across a hole or asymptote. Finally, use a calculator only as a check, since window settings can hide holes, flatten steep sections, or make distant behavior misleading.