Curve Sketching

In The Slope Detective, your group was handed a graph of $f^{\prime }$ — only the derivative, never the function — and asked to reconstruct what $f$ must look like. The startling part was how much you could recover: every climb, every turn, every flattening, all legible in a graph of slopes. And the honest limit of the method mattered just as much: your group’s $f$ and the next group’s $f$ differed by a vertical shift, and both were right. The derivative knows a function’s shape completely and its height not at all — infinitely many correct answers, one shape.

What the first derivative knows

The sign of $f^{\prime }$ is the story of $f$‘s direction:

• $f^{\prime }(x)>0$ on an interval — $f$ is increasing there.
• $f^{\prime }(x)<0$ — decreasing.
• $f^{\prime }(x)=0$ — a flat moment: possibly a local maximum, possibly a local minimum, possibly a pause mid-climb.

That last “possibly” is why detectives check both sides. A local maximum is $f^{\prime }$ changing from $+$ to $-$; a minimum is $-$ to $+$; no sign change, no turn — $f(x)=x^4-4x^3$ has $f^{\prime }(0)=0$ and sails straight through.

What the second derivative adds

$f^{\prime \prime }$ is the derivative’s derivative — the same idea that was acceleration in Motion on a Line, now read as bending. Where $f^{\prime \prime }>0$ the slopes are increasing and the graph is concave up, holding water; where $f^{\prime \prime }<0$ it is concave down, shedding it. A point where the concavity actually changes is a point of inflection — the graph’s wrist-flick, where tangent slopes stop growing and start shrinking or the reverse. At an inflection point $f^{\prime \prime }=0$, and $f^{\prime }$ itself has a local maximum or minimum: the steepest moment of the climb.

The sketching checklist

Consolidating from the boards, the full routine for a polynomial like $f(x)=x^3-6x^2+9x$:

• Intercepts: factor if you can — $x(x-3{)}^2$ has roots at 0 and 3.
• Compute $f^{\prime }$ and find its zeros: $3(x-1)(x-3)$, so 1 and 3.
• Sign chart for $f^{\prime }$: increasing, then decreasing, then increasing — a local maximum at $(1,4)$, a local minimum at $(3,0)$.
• Compute $f^{\prime \prime }$ and its zeros: $6x-12$, so an inflection point at $(2,2)$ — concave down before, up after.
• Sketch, then verify with technology. Agreement is the point; a disagreement is data about where your chart went wrong.

Notice the double root at $x=3$ and the local minimum at $(3,0)$ are the same fact seen twice — the graph touching the axis without crossing. When two lines of evidence corroborate, the sketch is solid. Curve Sketching Practice runs this routine until it is yours, and a worked sketch makes an excellent notes-to-future-self entry in your Math Journal. The same machinery, pointed at “which value is best?” instead of “what does it look like?”, is Optimization.

Curriculum connection

B1.1

sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes) Sample problem: Investigate the effect on the graph of the derivative of applying vertical and horizontal translations to the graph of a given function.

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B1.3

determine algebraically the equation of the second derivative $f^{\prime \prime }(x)$ of a polynomial or simple rational function $f(x)$, and make connections, through investigation using technology, between the key features of the graph of the function (e.g., increasing/decreasing intervals, local maxima and minima, points of inflection, intervals of concavity) and corresponding features of the graphs of its first and second derivatives (e.g., for an increasing interval of the function, the first derivative is positive; for a point of inflection of the function, the slopes of tangents change their behaviour from increasing to decreasing or from decreasing to increasing, the first derivative has a maximum or minimum, and the second derivative is zero) Sample problem: Investigate, using graphing technology, connections between key properties, such as increasing/decreasing intervals, local maxima and minima, points of inflection, and intervals of concavity, of the functions $f(x)=4x+1$, $f(x)=x^2+3x-10$, $f(x)=x^3+2x^2-3x$, and $f(x)=x^4+4x^3-3x^2-18x$ and the graphs of their first and second derivatives.

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B1.4

describe key features of a polynomial function, given information about its first and/or second derivatives (e.g., the graph of a derivative, the sign of a derivative over specific intervals, the x-intercepts of a derivative), sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible Sample problem: The following is the graph of the function $g(x)$. [The document prints the graph of y = g(x) here.] If $g(x)$ is the derivative of $f(x)$, and $f(0)=0$, sketch the graph of $f(x)$. If you are now given the function equation $g(x)=(x-1)(x-3)$, determine the equation of $f^{\prime \prime }(x)$ and describe some features of the equation of $f(x)$. How would $f(x)$ change graphically and algebraically if $f(0)=2$?

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B1.5

sketch the graph of a polynomial function, given its equation, by using a variety of strategies (e.g., using the sign of the first derivative; using the sign of the second derivative; identifying even or odd functions) to determine its key features (e.g., increasing/decreasing intervals, intercepts, local maxima and minima, points of inflection, intervals of concavity), and verify using technology

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