Curve Sketching Reference Cheat Sheet

A printable reference covering first derivatives, second derivatives, critical points, concavity, asymptotes, and curve sketching workflow for grades 11-12.

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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 $f'(x)$ tells where a function is increasing or decreasing and helps locate local extrema. The second derivative $f''(x)$ 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 $f'(x)=0$ or where $f'(x)$ is undefined, as long as $x$ is in the domain of $f$ .
If $f'(x)>0$ on an interval, then $f(x)$ is increasing on that interval.
If $f'(x)<0$ on an interval, then $f(x)$ is decreasing on that interval.
A local maximum can occur where $f'(x)$ changes from positive to negative, and a local minimum can occur where $f'(x)$ changes from negative to positive.
If $f''(x)>0$ on an interval, then $f(x)$ is concave up on that interval.
If $f''(x)<0$ on an interval, then $f(x)$ is concave down on that interval.
A possible inflection point occurs where $f''(x)=0$ or $f''(x)$ is undefined, but the concavity must change there.
A horizontal asymptote can be found by evaluating $\lim_{x\to\infty} f(x)$ and $\lim_{x\to -\infty} f(x)$ when those limits are finite.

Vocabulary

Critical Number: A value $x=c$ in the domain of $f$ where $f'(c)=0$ or $f'(c)$ does not exist.
Increasing Interval: An interval where the function values rise as $x$ increases, usually shown by $f'(x)>0$ .
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 $f''(x)>0$ or $f''(x)<0$ .
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 $f'(x)=0$ as a maximum or minimum is wrong because the derivative must change sign or another test must confirm the extremum.
Forgetting points where $f'(x)$ is undefined is wrong because corners, cusps, or vertical tangents can also create critical numbers.
Calling every solution of $f''(x)=0$ an inflection point is wrong because concavity must actually change across that value.
Ignoring the domain when finding asymptotes is wrong because excluded $x$ -values and discontinuities control where vertical asymptotes may occur.
Sketching before making a sign chart is wrong because the signs of $f'(x)$ and $f''(x)$ determine increasing intervals, decreasing intervals, and concavity.

Practice Questions

1 For $f(x)=x^3-3x^2-9x+1$ , find the critical numbers and determine where $f(x)$ is increasing or decreasing.
2 For $f(x)=x^4-4x^2$ , find the intervals of concavity and all inflection points.
3 For $f(x)=\frac{x^2+1}{x-2}$ , identify the vertical asymptote and determine whether the function has a slant asymptote.
4 Explain why a point where $f''(x)=0$ is not automatically an inflection point.

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