B. Derivatives and Their Applications · Curve Sketching: Connecting f, f′ and f″

Curve Sketching I: Polynomial Functions

B1.5About 75 minutes

Learning goals

I can follow a complete curve-sketching routine for a polynomial function.
I can use symmetry (even and odd functions), intercepts and end behaviour along with f′ and f″.
I can produce an accurate sketch with exact key points and verify it with technology.

Words to know

even function
f(−x) = f(x) for all x; the graph is symmetric about the y-axis.

odd function
f(−x) = −f(x) for all x; the graph is symmetric about the origin (unchanged by a 180° rotation).

end behaviour
What f(x) does as x → ∞ and x → −∞; for a polynomial it is decided by the leading term.

order of a zero
The number of times its factor repeats; odd order crosses the x-axis, even order touches it.

Putting it all together

In MHF4U (Polynomial Functions and Equations) you sketched polynomials from their zeros, the order of each zero, and their end behaviour. Those sketches had the right overall shape, but you had to guess where the turning points were and how the graph bent. Now you can find every key feature exactly.

Graphing technology draws a curve in a second. So why learn to sketch by hand? Because a screen shows only a window. Calculus tells you what is outside the window, guarantees you have not missed a turning point, and gives exact values like 6√3 instead of 10.392. Engineers, economists and data scientists use this reasoning to check that computer output makes sense.

Four stages. Gather all the facts first, then draw. The sign chart for f′ and f″ is the backbone of every sketch.

The full checklist

Domain: all real numbers for a polynomial.
Intercepts: y-intercept f(0); x-intercepts from factoring (note the order of each zero: odd order crosses, even order touches).
Symmetry: if f(−x) = f(x), the graph is even (symmetric about the y-axis); if f(−x) = −f(x), it is odd (symmetric about the origin). For polynomials: only even powers → even; only odd powers → odd.
End behaviour: from the leading term.
f′: critical numbers, intervals of increase/decrease, local extrema (with y-values).
f″: intervals of concavity, inflection points (with y-values).
Sketch: plot all key points, then draw each interval with the right rise/fall and bend.
Verify with graphing technology (Desmos, GeoGebra or a graphing calculator).

Worked example

Example 1: A cubic

Sketch f(x) = x³ − 3x² using the checklist.

Intercepts: f(0) = 0. f(x) = x²(x − 3): zeros x = 0 (order 2, the graph touches) and x = 3 (order 1, it crosses).
Symmetry: both odd and even powers appear, so neither even nor odd.
End behaviour: leading term x³: f → −∞ as x → −∞ and f → ∞ as x → ∞.
f′(x) = 3x² − 6x = 3x(x − 2). Signs +, −, +. Local max at (0, 0); local min at (2, f(2)) = (2, 8 − 12) = (2, −4).
f″(x) = 6x − 6. Concave down on (−∞, 1), concave up on (1, ∞). IP at (1, f(1)) = (1, −2).
Sketch: plot (0, 0), (1, −2), (2, −4), (3, 0). Draw rising and concave down up to (0, 0), falling and concave down to (1, −2), falling and concave up to (2, −4), then rising and concave up through (3, 0).

See the graph below. Summary: max (0, 0), min (2, −4), IP (1, −2), zeros 0 and 3.

interval	x < 0	0 < x < 1	1 < x < 2	x > 2
f′	+	−	−	+
f″	−	−	+	+
shape of f	rising, ∩	falling, ∩	falling, ∪	rising, ∪

The combined chart for f(x) = x³ − 3x². Each column tells you exactly how to draw that piece.

The finished sketch: a double zero at the origin (the graph touches the axis there, at the local maximum), a local minimum at (2, −4), an inflection point at (1, −2), and a simple zero at x = 3.

Worked example

Example 2: An odd quintic

Sketch f(x) = x⁵ − 5x³.

Intercepts: f(x) = x³(x² − 5): zeros x = 0 (order 3: crosses, flattening out) and x = ±√5 ≈ ±2.24. y-intercept 0.
Symmetry: only odd powers, so f is odd: f(−x) = −f(x). Whatever happens at x happens upside down at −x.
End behaviour: like x⁵: down on the left, up on the right.
f′(x) = 5x⁴ − 15x² = 5x²(x² − 3). Critical numbers 0, ±√3. 5x² ≥ 0, so f′ has the sign of x² − 3: + for |x| > √3, − for |x| < √3 (x ≠ 0). Local max at x = −√3, local min at x = √3, no extremum at 0.
Values: x = √3 gives x² = 3, so f(√3) = x³(x² − 5) = 3√3(−2) = −6√3 ≈ −10.39. By symmetry f(−√3) = 6√3.
f″(x) = 20x³ − 30x = 10x(2x² − 3). Zeros 0 and ±√(3/2) ≈ ±1.22. Signs: −, +, −, +. All three are inflection points.
Values: x² = 3/2 gives f = x³(x² − 5) = x(3/2)(−7/2) = −(21/4)x. So f(√1.5) ≈ −6.43 and f(−√1.5) ≈ 6.43; f(0) = 0.
Sketch with the chart: rising ∩ up to the max, falling ∩ to the IP near (−1.22, 6.43), falling ∪ to the flat IP at the origin, falling ∩ to the IP near (1.22, −6.43), falling ∪ to the min, rising ∪ after.

Max (−√3, 6√3), min (√3, −6√3), IPs (−√1.5, 21√1.5/4), (0, 0), (√1.5, −21√1.5/4); zeros 0, ±√5.

An odd function: rotating the graph 180° about the origin gives the same graph. So the maximum (−√3, 6√3 ≈ 10.4) matches the minimum (√3, −6√3), and the inflection points come in a matching pair (about (±1.22, ∓6.43)) plus the horizontal one at the origin.

Verifying with technology

After sketching, type the function into a free graphing tool such as Desmos or GeoGebra (or use a graphing calculator). Check that:

the turning points and inflection points are where you calculated;
the window shows all of them (adjust it using your y-values, for example y from −12 to 12 for Example 2);
the intercepts and end behaviour match.

If something disagrees, the error is almost always in factoring f′ or f″, or in substituting into f′ instead of f.

Just for fun

Funny but true: before computers, drafters drew smooth curves through plotted points with a plastic template called a French curve. Calculus would have told them exactly where to bend it.

Hands-on activity

Sketch first, then check

You need: graph paper, a pencil, an eraser, and a device with a free graphing tool (Desmos or GeoGebra in a browser) or a graphing calculator.

Choose two functions: f(x) = −x³ + 3x + 2 and g(x) = x⁴ − 4x².
Sketch each by hand using the full checklist. Label every key point with exact coordinates.
Only then graph them on the device. Zoom so the window shows all key points.
Score yourself: one point for each correctly placed intercept, extremum and inflection point.

What to look for: f has a min at (−1, 0), max at (1, 4), IP at (0, 2), zeros −1 (double) and 2 (since −x³ + 3x + 2 = −(x + 1)²(x − 2)). g has mins at (±√2, −4), max at (0, 0), IPs at (±√(2/3), −20/9), zeros 0 (double) and ±2.

Your turn

For f(x) = x³ − 12x, give the intercepts, symmetry and end behaviour.

For f(x) = x³ − 12x, find the extrema and inflection point.

Is f(x) = x⁴ − 3x² + 1 even, odd or neither?

Is f(x) = x³ + x² odd?

Sketch, in words, f(x) = −x³ + 3x² (find all key points).

Write the combined chart (signs of f′, f″) for f(x) = x³ − 3x on the intervals x < −1, −1 < x < 0, 0 < x < 1, x > 1.

More practice: warm-up

Which test gives symmetry about the y-axis?

Zero of order 2: does the graph cross or touch?

End behaviour of f(x) = −2x⁴ + x?

y-intercept of f(x) = x³ − 4x² + 7?

Is f(x) = x⁵ − x odd?

More practice: on your own

Sketch f(x) = 2x³ + 3x² − 12x, listing all key points.

Sketch f(x) = x⁴ − 4x³, listing all key points.

Sketch f(x) = x³ + 3x² + 3x + 1. (Hint: factor.)

Sketch the odd function f(x) = 3x − x³.

Use symmetry: f is even, has a local max at (0, 3) and a local min at (2, −1). What other key point must it have?

Challenge

Sketch f(x) = x³ − 3x + 1 even though its zeros do not factor nicely. Show there are three zeros and locate each between consecutive integers.

Prove that an odd function defined at x = 0 passes through the origin. Then show that every odd polynomial has f″(0) = 0.

For which k does f(x) = x⁴ + kx³ have an inflection point at x = 1? Then sketch it.

Check your understanding

Quiz

Which function is odd?

For f(x) = x³ − 3x², where is the local minimum?

On an interval, f′ < 0 and f″ < 0. How should you draw that piece?

An even polynomial has a local minimum at (3, −5). Which point must also be a local minimum?

Why is sketching by hand useful even with graphing technology?

For f(x) = x⁵ − 5x³, how many points of inflection are there?

For parents and teachers

Big idea (B1.5): a full curve sketch combines MHF4U facts (intercepts, symmetry, end behaviour) with calculus facts (extrema from f′, concavity and inflection points from f″). A combined sign chart turns all of this into a clear set of instructions for drawing each piece.

How to help: ask your teen to talk you through the checklist for one function before drawing. Then let them check with a free graphing tool. The habit of predicting before checking is what university courses expect.

Common difficulties:

Skipping the y-values of key points, so the sketch has the right shape but the wrong height.
Drawing the pieces with the wrong bend (rising ∪ vs rising ∩).
Forgetting that a zero of order 3 flattens as it crosses.
Using a graphing window that hides features.