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 $y = -1$.
• $\frac{1}{f}$ 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.
• $f^2 \ge 0$ 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.

Build the reciprocal graph, stage by stage

The reciprocal $y = \frac{1}{f}$ is the most-examined combination, and the safest way to sketch it is to mark the structure from $f$ first, then draw. Below, $y = \frac{1}{x^2 - 1}$ is built from $f(x) = x^2 - 1$ one decision at a time. (The 2022 HSC asked exactly this kind of "describe the features of $\frac{1}{f}$" question.)

Stage 1, plot the parent function. Draw $y = f(x) = x^2 - 1$, an upward parabola. Mark the two features that drive the reciprocal: the zeros at $x = -1$ and $x = 1$, and the minimum at $(0, -1)$. Everything about $\frac{1}{f}$ follows from these.

Stage 2, turn zeros into asymptotes and mark the meeting lines. Each zero of $f$ (at $x = \pm 1$) becomes a vertical asymptote of $\frac{1}{f}$, because dividing by zero blows up. Draw those dashed verticals. Also draw the lines $y = 1$ and $y = -1$: wherever $f = \pm 1$, the reciprocal equals $\pm 1$ too, so $\frac{1}{f}$ and $f$ cross on these lines.

Stage 3, sketch the three branches. Now draw $\frac{1}{f}$ between and outside the asymptotes. Outside $(\,x < -1$ and $x > 1)$ the parabola is positive and large, so the reciprocal is positive and small, hugging the $x$-axis far out and rising to $+\infty$ at each asymptote. Between the asymptotes $f$ is negative (down to $-1$ at the centre), so $\frac{1}{f}$ is negative, plunging to $-\infty$ near each asymptote.

Stage 4, check the extremum flips. The minimum of $f$ at $(0, -1)$, where $f$ is negative, becomes a local maximum of $\frac{1}{f}$ at the same point $(0, -1)$: $\frac{1}{-1} = -1$, and reciprocating a "most negative" value gives the "least negative" one. This sign-aware flip of turning points is the detail markers look for.

How exam questions ask about combining functions

The combinations come up in a handful of standard phrasings:

• "Describe the key features of $y = \frac{1}{f(x)}$" given the zeros and sign of $f$. Zeros become vertical asymptotes, the sign is preserved, the curves meet at $y = \pm 1$, and turning points flip (max becomes min where $f > 0$). State each (2022 HSC Q14).
• "Sketch $y = f(x)\,g(x)$" with one factor a sine or cosine. Mark zeros (the union of both factors' zeros), draw the envelope $y = \pm f$, and show the oscillation with growing or decaying amplitude (2021 HSC Q15).
• "Sketch $y = f(x) + g(x)$." Add ordinates: at each $x$, add the two heights. Crossings of the $x$-axis occur where $f = -g$, not where either is zero.
• "Sketch $y = \frac{f(x)}{g(x)}$." Zeros of $f$ give $x$-intercepts; zeros of $g$ give vertical asymptotes; a common zero may be a hole; compare growth rates for any horizontal asymptote.
• "Sketch $y = [f(x)]^2$." Fold the graph above the $x$-axis; the zeros of $f$ become double-root touches, and $y$ never goes negative.
• "On the diagram of $y = f(x)$, sketch $y = \frac{1}{f(x)}$" (a build-on-the-given-graph task). Read the zeros, the $\pm 1$ crossings and the turning points straight off the printed curve and apply the four reciprocal rules.