MCV4U Sketching Polynomial and Simple Rational Functions: Notes and Practice

Graphing software draws a curve in a second, so why sketch by hand? Because a screen only shows the window you chose. A local maximum just outside the window, two turning points squeezed close together, or a gentle inflection point can all hide in a picture. Calculus finds EVERY critical number and every candidate inflection point by solving equations, so a sketch built from $f^{'}$ and $f^{''}$ , with those equations solved completely, shows all of the shape. Technology then becomes a check rather than a guess.

Fig 1A curve-sketching flowchart: domain, then intercepts, then symmetry (even or odd), then the first derivative for critical points, increasing and decreasing intervals and local extrema, then the second derivative for concavity and inflection points, then asymptotes for rational functions, and finally sketch and verify.

DOMAIN: polynomials are defined for all real $x$ . For a rational function, exclude the zeros of the denominator.
INTERCEPTS: the $y$ -intercept is $f (0)$ ; the $x$ -intercepts are the solutions of $f (x) = 0$ (factor if you can).
SYMMETRY: check whether $f (- x) = f (x)$ (even) or $f (- x) = - f (x)$ (odd). Symmetry halves the work.
FIRST DERIVATIVE: critical numbers, intervals of increase and decrease, local maximums and minimums.
SECOND DERIVATIVE: intervals of concavity and points of inflection.
ASYMPTOTES AND END BEHAVIOUR: vertical and horizontal asymptotes for rational functions; for a polynomial, the leading term decides what happens as $x \to \pm \infty$ .
SKETCH: plot the key points, follow the sign charts between them, then verify with graphing technology.

Symmetry: even and odd functions

A function is EVEN if $f (- x) = f (x)$ for every $x$ in its domain; its graph is symmetric about the $y$ -axis. A polynomial whose terms all have even powers (a constant counts as $x^{0}$ ) is even, such as $x^{4} - 8 x^{2} + 7$ . A function is ODD if $f (- x) = - f (x)$ ; its graph has rotational symmetry about the origin (a half turn maps it onto itself). A polynomial with only odd powers and no constant term is odd, such as $x^{3} - 12 x$ . Most functions are neither. Symmetry carries over to features: if an even function has a minimum at $(2, - 9)$ , it also has one at $(- 2, - 9)$ ; if an odd function has a maximum at $(- 2, 16)$ , it has a minimum at $(2, - 16)$ .

Feature	Result for f(x) = x⁴ − 8x² + 7
Symmetry	Even: only even powers, so symmetric about the $y$ -axis
Intercepts	$y$ -intercept $7$ ; $f (x) = (x^{2} - 1) (x^{2} - 7)$ , so $x = \pm 1$ and $x = \pm 7 \approx \pm 2.65$
First derivative	$f^{'} (x) = 4 x^{3} - 16 x = 4 x (x - 2) (x + 2);$ signs $-, +, -, +$
Extrema	Local minimums $(\pm 2, - 9)$ (both absolute); local maximum $(0, 7)$
Second derivative	$f^{''} (x) = 12 x^{2} - 16$ , zero at $x = \pm \frac{2 3}{3} \approx \pm 1.15$ ; signs $+, -, +$
Concavity	Up on $(- \infty, - \frac{2 3}{3})$ and $(\frac{2 3}{3}, \infty)$ , down between
Inflection points	$(\pm \frac{2 3}{3}, - \frac{17}{9}) \approx (\pm 1.15, - 1.89)$
End behaviour	Leading term $x^{4}$ : $f (x) \to \infty$ as $x \to \pm \infty$ (a W shape)

A complete feature list for an even quartic. The right half of the graph is mirrored onto the left half.

How many bumps can a polynomial have

A polynomial of degree

n

has a derivative of degree

n - 1

, so it has at most

n - 1

critical numbers, hence at most

n - 1

turning points, and at most

n - 2

points of inflection. A cubic has at most

2

turning points and exactly

1

inflection point; a quartic has at most

3

turning points and at most

2

inflection points. If your sketch has more, something went wrong.

VideoJensenMath sketches polynomial functions by combining intervals of increase and decrease from f′ with intervals of concavity from f″, following the same checklist as this lesson.Watch on YouTube

Simple rational functions

Rational functions add two features you met in Advanced Functions. A VERTICAL ASYMPTOTE occurs at a zero of the denominator that does not cancel with the numerator; the graph shoots off to $\pm \infty$ there. A HORIZONTAL ASYMPTOTE describes the end behaviour: if the degree of the numerator is less than the degree of the denominator, it is $y = 0$ ; if the degrees are equal, it is $y = \frac{a}{b}$ , where $a$ and $b$ are the leading coefficients of the numerator and the denominator. Find these first, then use $f^{'}$ and $f^{''}$ exactly as before.

Put the asymptotes on every sign chart

f^{'}

and

f^{''}

can change sign at a vertical asymptote without any maximum, minimum or inflection point there, because the function is not defined at that

x

. Include the vertical asymptotes as boundaries on your sign charts, but never list them as critical numbers or inflection points.

Example: $f (x) = \frac{1}{x ^{2} - 4}$ . Domain $x \neq = \pm 2$ ; vertical asymptotes $x = \pm 2$ ; horizontal asymptote $y = 0$ (degree $0$ over degree $2$ ).
$f (- x) = f (x)$ , so it is even. The $y$ -intercept is $- \frac{1}{4}$ , and there are no $x$ -intercepts because the numerator is never $0$ .
$f^{'} (x) = \frac{- 2 x}{( x ^{2} - 4 ) ^{2}}$ : positive for $x < 0$ and negative for $x > 0$ (skipping $\pm 2$ ), so $f$ increases on $(- \infty, - 2)$ and $(- 2, 0)$ , decreases on $(0, 2)$ and $(2, \infty)$ , with a local maximum at $(0, - \frac{1}{4})$ .
$f^{''} (x) = \frac{6 x ^{2} + 8}{( x ^{2} - 4 ) ^{3}}$ : the numerator is always positive, so the sign follows the denominator. Concave up for $∣ x ∣ > 2$ , concave down for $- 2 < x < 2$ .
The concavity changes only at $x = \pm 2$ , where $f$ is undefined, so there are NO points of inflection.

Verify with technology

Graph the function in Desmos or on a graphing calculator, and also graph

f^{'}

and

f^{''}

. Check that every turning point you found sits where the graph of

f^{'}

crosses the

x

-axis, and every inflection point sits where the graph of

f^{''}

crosses. Zoom out to confirm the end behaviour and zoom in near close-together features.

Sketching Polynomial and Simple Rational Functions

Key things to remember

Symmetry: even and odd functions

Simple rational functions

Worked examples

Where marks get lost

Flashcards 7

Practice