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Subsection 3.6.5 A Checklist for Sketching

Above we have described how we can use our accumulated knowledge of derivatives to quickly identify the most important qualitative features of graphs \(y=f(x)\text{.}\) Here we give the reader a quick checklist of things to examine in order to produce an accurate sketch based on properties that are easily read off from \(f(x)\text{,}\) \(f'(x)\) and \(f''(x)\text{.}\)

Subsubsection 3.6.5.1 A Sketching Checklist

  1. Features of \(y = f(x)\) that are read off of \(f(x)\text{:}\)

    • First check where \(f(x)\) is defined. Then
    • \(y=f(x)\) is plotted only for \(x\)'s in the domain of \(f(x)\text{,}\) i.e. where \(f(x)\) is defined.
    • \(y = f(x)\) has vertical asymptotes at the points where \(f(x)\) blows up to \(\pm\infty\text{.}\)
    • Next determine whether the function is even, odd, or periodic.
    • \(y=f(x)\) is first plotted for \(x\ge 0\) if the function is even or odd. The rest of the sketch is then created by reflections.
    • \(y=f(x)\) is first plotted for a single period if the function is periodic. The rest of the sketch is then created by translations.
    • Next compute \(f(0)\text{,}\) \(\lim_{x\rightarrow\infty} f(x)\) and \(\lim_{x\rightarrow-\infty} f(x)\) and look for solutions to \(f(x)=0\) that you can easily find. Then
    • \(y = f(x)\) has \(y\)–intercept \(\big(0, f(0)\big)\text{.}\)
    • \(y = f(x)\) has \(x\)–intercept \((a,0)\) whenever \(f(a)=0\)
    • \(y = f(x)\) has horizontal asymptote \(y=Y\) if \(\lim_{x\rightarrow\infty} f(x)=L\) or \(\lim_{x\rightarrow-\infty} f(x)=L\text{.}\)
  2. Features of \(y=f(x)\) that are read off of \(f'(x)\text{:}\)

    • Compute \(f'(x)\) and determine its critical points and singular points, then
    • \(y=f(x)\) has a horizontal tangent at the points where \(f'(x)=0\text{.}\)
    • \(y=f(x)\) is increasing at points where \(f'(x) \gt 0\text{.}\)
    • \(y=f(x)\) is decreasing at points where \(f'(x) \lt 0\text{.}\)
    • \(y=f(x)\) has vertical tangents or vertical asymptotes at the points where \(f'(x)=\pm\infty\text{.}\)
  3. Features of \(y=f(x)\) that are read off of \(f''(x)\text{:}\)

    • Compute \(f''(x)\) and determine where \(f''(x)=0\) or does not exist, then
    • \(y=f(x)\) is concave up at points where \(f''(x) \gt 0\text{.}\)
    • \(y=f(x)\) is concave down at points where \(f''(x) \lt 0\text{.}\)
    • \(y=f(x)\) may or may not have inflection points where \(f''(x)=0\text{.}\)