Sketch graphs of sums, differences, products, quotients, squares and reciprocals of two known functions

What this dot point is asking

NESA wants you to sketch a function built by combining two known functions. The combinations are sums $f + g$, differences $f - g$, products $f g$, quotients $\frac{f}{g}$, squares $f^2$, and reciprocals $\frac{1}{f}$. You need to read features off the parent graphs and predict the combined graph.

The answer

Sums and differences

For $y = f(x) + g(x)$, add the heights of the two graphs at each $x$. Useful checks:

• Zeros of $f + g$ occur where the two graphs are reflections of each other through the $x$-axis, that is $f(x) = -g(x)$, not generally where either function is zero alone.
• If $f$ is bounded and $g$ is unbounded, the long-term behaviour of $f + g$ is the same as $g$.
• For $f - g$, subtract the heights. At points where $f = g$, the difference is zero.

Products

For $y = f(x) g(x)$, multiply the heights:

• Zeros of $f g$ are the union of the zeros of $f$ and $g$.
• The sign of $f g$ follows the rule of signs: $(+)(+) = +$, $(+)(-) = -$, and so on.
• If one factor is bounded between $-1$ and $1$ (like $\sin x$), the other factor acts as an envelope: $|f g| \le |f|$ and the graph oscillates between $y = \pm f(x)$.
• If both factors grow, $f g$ grows faster.

Quotients

For $y = \frac{f(x)}{g(x)}$:

• Zeros of $\frac{f}{g}$ are the zeros of $f$, provided $g$ is non-zero there.
• Vertical asymptotes occur at zeros of $g$ where $f$ is non-zero.
• If $g(x) = 0$ and $f(x) = 0$ at the same point, there is a hole or a finite limit; check carefully.
• Horizontal asymptotes come from the ratio of long-term behaviours: $\frac{f}{g} \to 0$ if $g$ grows faster, a non-zero constant if they grow at the same rate, and $\pm \infty$ if $f$ grows faster.

Reciprocals

The graph of $y = \frac{1}{f(x)}$ comes from $y = f(x)$ by these rules:

• Where $f(x) = 0$, $\frac{1}{f}$ has a vertical asymptote.
• Where $f(x) = \pm 1$, the reciprocal also equals $\pm 1$, so the two graphs meet on the lines $y = 1$ and $y = -1$.
• IMATH_50 has the same sign as $f$ everywhere $f \neq 0$.
• Local maxima of $f$ where $f > 0$ become local minima of $\frac{1}{f}$, and vice versa, because reciprocating flips relative size.
• As $f(x) \to \pm \infty$, $\frac{1}{f(x)} \to 0$. As $f(x) \to 0^{\pm}$, $\frac{1}{f(x)} \to \pm \infty$.

Squares

For $y = (f(x))^2$:

• Zeros of $f^2$ are the zeros of $f$, but now they are double roots: the graph touches the $x$-axis and turns.
• IMATH_64 everywhere, so the graph never dips below the $x$-axis.
• Where $f(x) = \pm 1$, $f^2(x) = 1$. Where $|f(x)| > 1$, $f^2 > |f|$. Where $|f(x)| < 1$, $f^2 < |f|$.
• Extrema of $f^2$ occur where $f f' = 0$, that is where $f = 0$ or where $f$ has an extremum.

Worked examples

Reciprocal of IMATH_76

$y = \csc x = \frac{1}{\sin x}$ has vertical asymptotes at $x = k \pi$ (zeros of $\sin x$), agrees with $\sin x$ at $\sin x = \pm 1$ (so at $x = \frac{\pi}{2} + k \pi$), and is positive on intervals where $\sin x > 0$.

Square of IMATH_84

$y = \sin^2 x$ has zeros at $x = k \pi$ (double roots), touches the $x$-axis there, is bounded above by $1$, and equals $1$ at $x = \frac{\pi}{2} + k \pi$. Its period is $\pi$, half that of $\sin x$.

Quotient with shared zero

$y = \frac{\sin x}{x}$ has a hole at $x = 0$ (because both numerator and denominator are zero there) with limit $1$. Elsewhere it inherits zeros from $\sin x$ at $x = \pm \pi, \pm 2 \pi, \dots$ and decays in amplitude like $\frac{1}{|x|}$ as $|x| \to \infty$.

Product with growing envelope

$y = e^{-x} \sin x$ for $x \ge 0$ oscillates with the same zeros as $\sin x$ (at $x = k \pi$), but the amplitude decays. The envelopes are $y = \pm e^{-x}$, and the graph touches them where $\sin x = \pm 1$.

Sum: line plus sine

$y = x + \sin x$ has the line $y = x$ as a "spine" with small oscillations of amplitude $1$ added. The graph is always within $1$ of $y = x$ and crosses the line at every multiple of $\pi$.

Common traps

Treating reciprocals like reflections. $\frac{1}{f}$ is not a reflection. The shape distorts: large values become small and vice versa.

Forgetting asymptotes are about zeros of the denominator. For $\frac{f}{g}$, vertical asymptotes come from $g(x) = 0$, not $f(x) = 0$. Zeros come from $f(x) = 0$ (with $g(x) \neq 0$).

Square has zeros, not just minimums. $f^2$ has zeros wherever $f$ does. The graph touches the $x$-axis at each one rather than crossing.

Misreading envelopes. For $y = f g$ with $|g| \le 1$, the envelope is $y = \pm f$, not $y = f$ alone. Both branches matter.

Missing holes versus asymptotes in quotients. If $f$ and $g$ both vanish at the same point with a common factor, you get a hole, not an asymptote. Factor and cancel before concluding.

In one sentence

Combine known graphs feature by feature: sums add heights, products multiply heights (with the smaller factor acting as an envelope), quotients put zeros of the numerator on the $x$-axis and zeros of the denominator at vertical asymptotes, reciprocals flip $0 \leftrightarrow \infty$ and meet at $y = \pm 1$, and squares fold everything above the $x$-axis with double-root touches at the original zeros.