Mathematics, four units · Grade 11

What is the order of operations in a full investigation of a polynomial?

Domain, intersections with the axes, the derivative, setting the derivative to zero, classifying each stationary point, intervals of increase and decrease, and only then the sketch. The order is not a matter of style: each stage supplies the data for the next, and the sketch comes last so that it fits the results rather than determining them.

Learning objectives

The derivative as a slope

The derivative at a point is the slope of the tangent to the graph there. Positive means the function is rising, negative means it is falling, and zero means the tangent is horizontal.

Differentiating a polynomial is one rule: multiply each term by its power and reduce the power by one. A constant differentiates to zero, because the graph of a constant is a horizontal line with slope zero.

It follows that stationary points sit wherever the derivative vanishes. Set the derivative equal to zero, solve, and you have the x values of the candidates.

Classify, do not merely locate

Finding a point where the derivative is zero is not the end of the answer. You must decide whether it is a maximum, a minimum or a horizontal inflection, and justify it.

The convenient method is a sign table: mark all the zeros of the derivative on a line and test the sign of the derivative in each interval between them. Plus to minus is a maximum; minus to plus is a minimum; the same sign on both sides is not a turning point at all.

You test by substituting one convenient number from each interval into the derivative, and only the sign matters, never the value. That is one line of work per interval.

From the table to the sketch

The intervals of increase and decrease are read straight off that same table: an interval where the derivative is positive is an interval of increase. Write them as ranges of x, not as a list of points.

The sketch comes last and is assembled from what you already have: the y-intercept, the x-intercepts, the turning points with their values, and the direction of travel in each interval.

One final check worth ten seconds: for a cubic with a positive leading coefficient, the left tail falls to minus infinity and the right tail rises. If your sketch does otherwise, something has gone wrong.

Worked examples

  1. Investigate f(x) equals x cubed minus 3x squared. Find the stationary points.

    1. The derivative: 3x squared minus 6x
    2. Set to zero: 3x times (x minus 2) equals zero
    3. x equals 0 or x equals 2
    4. Substitute back into the original function

    Answer: The points (0, 0) and (2, minus 4)

  2. Classify those two points.

    1. Test the sign to the left of 0, say at x equals minus 1: positive
    2. Between 0 and 2, say at x equals 1: negative
    3. To the right of 2, say at x equals 3: positive

    Answer: (0, 0) is a maximum, plus to minus; (2, minus 4) is a minimum, minus to plus

  3. What are the intervals of increase and decrease of that function?

    1. The derivative is positive when x is less than 0
    2. Negative between 0 and 2
    3. Positive when x is greater than 2

    Answer: Increasing for x below 0 and for x above 2, decreasing between 0 and 2

Common mistakes

Presenting a zero of the derivative as a turning point without testing
A vanishing derivative can also mark a horizontal inflection, where the function carries on in the same direction. Without a sign table there is no way to know, and the classification mark is lost.
Writing intervals of increase as a list of points
Increase is a property of a whole interval, not of a single point. The answer is required as an inequality in x, and that is the form that is marked.
Sketching first and calculating afterwards
A guessed sketch fixes the conclusion in advance and the working gets bent to match it. The reverse order is what lets you discover an error instead of concealing it.

What to remember

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