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

The Seven Core Steps in Order

The order of operations in a full investigation of a polynomial is fixed because each stage supplies the inputs for the next. Start by noting the domain, which for any polynomial is all real numbers. Next, determine the axis intercepts by setting x = 0 to get the y-intercept, and setting y = 0 to solve for the x-intercepts.

After finding the intercepts, differentiate the polynomial using the power rule. Set this derivative equal to zero to locate the x-coordinates of candidate stationary points. Substitute those x-coordinates back into the original function to get the complete coordinates of each point.

Next, set up a sign table around those x-values to classify each point as a local maximum, local minimum, or horizontal point of inflection. Read your intervals of increase and decrease directly from the signs in this table. Finally, sketch the graph using only these proven values.

Classifying Points with a Sign Table

A zero derivative merely marks a horizontal tangent line, not necessarily a peak or a valley. To classify each candidate, place the critical x-values on a number line, splitting the domain into test intervals. Pick one test value in each interval and evaluate its sign in the first derivative.

If the derivative switches from positive to negative, the function shifts from rising to falling, creating a local maximum. If it switches from negative to positive, you have found a local minimum. If the sign remains the same on both sides, the point is a horizontal inflection point.

A Worked Walkthrough: f(x) = x³ - 3x²

Take f(x) = x³ - 3x². Its derivative is f'(x) = 3x² - 6x. Factoring gives 3x(x - 2) = 0, yielding candidate points at x = 0 and x = 2. Plug these into the original function to find the points: (0, 0) and (2, -4).

Test intervals for the derivative: at x = -1, f'(-1) = 9 (positive). At x = 1, f'(1) = -3 (negative). At x = 3, f'(3) = 9 (positive). Thus, (0, 0) is a local maximum and (2, -4) is a local minimum. The function increases on x < 0 and x > 2, and decreases on 0 < x < 2. Plot these coordinates, match the intervals, and draw the curve.

Common Mistakes to Avoid

Never sketch the curve first and force your calculations to match it. Preconceived curves lead to overlooking algebraic sign errors. Sketch strictly after all coordinates and signs are settled.

Remember that intervals of increase and decrease must be stated as inequalities of x, such as 0 < x < 2, rather than single coordinates or y-values.

Always evaluate test points in the first derivative f'(x) to check slopes, but substitute critical points into the original f(x) to locate the actual coordinates on the graph.