Calculus I · Unit 2B · lesson
Graph Analysis as a Coherent Story
Learning objectives
Organize critical numbers, sign charts, concavity, asymptotes, and endpoints into one analysis rather than a disconnected checklist.
From Local Slopes to the Shape of an Entire Graph
Before the formulas
Graph analysis in Graph Analysis as a Coherent Story is a coherent reconstruction problem. Domain, intercepts, limits, derivative signs, critical points, concavity, and asymptotes constrain the same curve. Build the picture in layers rather than trying to sketch from the original formula at once.
Keep a feature table. Each row should state the calculation, the interval or point, and the graphical consequence. This makes the final sketch a summary of evidence instead of an artistic guess.
Derivative tests turn formulas into a narrative
A complete graph analysis tells when the function rises, falls, bends upward, bends downward, and reaches turning points. It is a story assembled from signs of and , not a list of disconnected calculations.
Sign charts provide the grammar of that story. They organize intervals and make it harder to confuse a zero of the derivative with a zero of the original function.
A complete graph analysis combines information from several sources. The domain identifies where the graph can exist. Limits describe boundary and asymptotic behavior. Zeros locate intercepts. The sign of tells where the graph rises or falls. The sign of tells how the slopes change. Endpoints and critical numbers provide candidates for extrema.
The pieces should agree. If a sign chart says , your sketch must rise. If , its slopes must become less positive or more negative as increases. A sketch is therefore a visual summary of proved behavior, not an artistic guess with coordinate axes.
Read derivative signs as verbs
: rising. : falling. : slopes increasing. : slopes decreasing.
After the explanation
Use the section idea
Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.
Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.
List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.
Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.
Can every turn, bend, endpoint result, and asymptote in your sketch be traced to algebraic evidence?
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