Sketch the graphs of power functions $y = x^n$ and contrast even and odd powers, sketch a cubic or higher polynomial that is factored into linear factors using a sign table, sketch circles $x^2 + y^2 = r^2$ centred at the origin and recognise shifted circles, and sketch the rectangular hyperbola $y = \frac{1}{x}$ with its asymptotes

What this dot point is asking

So far the only curved graph you can sketch completely is the parabola. This dot point widens that toolkit to the next family of graphs the HSC expects you to sketch by hand from their equation: the power functions $y = x^n$, the cubics and higher polynomials that come already factored into linear factors, the circle $x^2 + y^2 = r^2$, and the rectangular hyperbola $y = \dfrac{1}{x}$. None of these needs calculus. Each is sketched from a small number of features (zeros, intercepts, sign, symmetry, asymptotes) plus a clear picture of the basic shape.

The unifying idea is that a graph changes from being above the $x$-axis to below it only where it crosses or touches the axis, that is, only at a zero. Between consecutive zeros the sign of $y$ cannot change, so a single test value tells you the sign of the whole interval. That one observation turns a factored polynomial of any degree into a sketch: find the zeros, test the sign in each gap, and join the dots with the right end behaviour. The circle and the hyperbola are then two more standard shapes worth knowing on sight. The marks in an exam come from getting the intercepts and the shape right, and from knowing the limits of the method, chiefly that the exact turning points of a cubic are a Year 12 (calculus) result and must not be invented.

The answer

Powers of x: even versus odd

The simplest non-linear graphs are the power functions $y = x^n$. Their shape depends almost entirely on whether the power $n$ is even or odd, because that decides what happens to a negative input. A negative number raised to an even power is positive, while raised to an odd power it stays negative: $(-2)^2 = 4$ but $(-2)^3 = -8$.

For an even power ($y = x^2$, $y = x^4$, $y = x^6$, ...) the graph is a U-shape that sits on or above the $x$-axis, touching it only at the origin, and is symmetric about the $y$-axis (since $(-x)^n = x^n$ when $n$ is even). For an odd power ($y = x^3$, $y = x^5$, ...) the graph rises from bottom-left to top-right, passing through the origin, and has point symmetry about the origin (since $(-x)^n = -x^n$ when $n$ is odd). Every one of these curves passes through $(0, 0)$ and $(1, 1)$; the even ones also pass through $(-1, 1)$, the odd ones through $(-1, -1)$. As the power increases, the curve becomes flatter near the origin (because a number between $-1$ and $1$ raised to a higher power is smaller, for example $0.5^4 = 0.0625$) and steeper further out.

Sketching a factored cubic with a sign table

A cubic is a polynomial of degree $3$, such as $y = x^3 - 2x^2 - 5x + 6$. You cannot sketch a general cubic by hand at this stage, but if it is factored into linear factors you can, because the factors hand you the zeros straight away. The graph can only change sign at a zero, so a sign table built by testing one value in each interval between the zeros tells you exactly where the curve is above the axis and where it is below.

Take $y = (x + 2)(x - 1)(x - 3)$. The factors are zero at $x = -2$, $x = 1$ and $x = 3$, so those are the three $x$-intercepts; the $y$-intercept is $y = (2)(-1)(-3) = 6$. Now choose a test value in each of the four intervals the zeros create and record the sign of $y$ (you only need the sign, so you can read it from the factors rather than multiply fully). The end behaviour comes for free: the leading term is $x^3$, so $y \to -\infty$ on the far left and $y \to +\infty$ on the far right, like every cubic with a positive leading coefficient. The four-panel build below turns this into a sketch.

Stage 1, the zeros and the $y$-intercept

Set each factor to zero to get the $x$-intercepts $x = -2$, $1$, $3$, and put $x = 0$ for the $y$-intercept $(0, 6)$. Plot these four points; they are the skeleton of the sketch.

Stage 2, the sign table

Dodge the zeros with a test value in each interval and record the sign of $y$. Here $f(-3) = -24$ (so $-$), $f(0) = 6$ (so $+$), $f(2) = -4$ (so $-$), and $f(4) = 18$ (so $+$). The signs across the four intervals are $-, +, -, +$.

Stage 3, plot the sign behaviour

Translate each sign into a position: where $y$ is positive the curve is above the $x$-axis, where $y$ is negative it is below. Sketch a rough run on the correct side of the axis in each interval (do not commit to a height yet).

Stage 4, the finished cubic

Join the runs through the three intercepts with a single smooth curve, matching the end behaviour (down on the left, up on the right). Mark the intercepts and the $y$-intercept only.

The same idea sketches a higher-degree polynomial that is factored into linear factors: find every zero, test the sign in each interval, and join with the correct end behaviour (for a positive leading coefficient, both ends go up if the degree is even, and the left goes down while the right goes up if the degree is odd). A repeated (squared) factor behaves differently at its zero: instead of crossing the axis, the curve touches it and turns back, because the squared factor does not change sign there. This is covered in the worked examples.

Circles centred at the origin

A circle is the set of points a fixed distance $r$ (the radius) from a fixed centre. For a circle centred at the origin, a point $(x, y)$ is on the circle exactly when its distance from $O$ is $r$. By Pythagoras (the distance formula), that distance squared is $x^2 + y^2$, so the equation is simply

To sketch it, read off $r$ as the square root of the number on the right, and draw a circle of that radius about the origin; it crosses the axes at $(\pm r, 0)$ and $(0, \pm r)$. For $x^2 + y^2 = 25$ the radius is $\sqrt{25} = 5$, so the circle cuts the axes at $\pm 5$. A point lies on the circle precisely when its coordinates satisfy the equation: $(3, 4)$ is on $x^2 + y^2 = 25$ because $3^2 + 4^2 = 9 + 16 = 25$.

The whole circle is a relation, not a function: a vertical line through the interior cuts it twice, so one $x$ can give two $y$ values. Solving for $y$ shows this directly: $y = \pm\sqrt{r^2 - x^2}$, two values for each $x$ strictly inside. Taking just the positive root $y = \sqrt{r^2 - x^2}$ keeps the upper semicircle (a function), and the negative root keeps the lower semicircle. A circle whose centre is shifted to $(h, k)$ has equation $(x - h)^2 + (y - k)^2 = r^2$: the same shape, moved $h$ across and $k$ up. For example $(x - 2)^2 + (y + 1)^2 = 9$ is a circle of radius $3$ centred at $(2, -1)$.

The rectangular hyperbola y = 1/x

The reciprocal function $y = \dfrac{1}{x}$ (equivalently $xy = 1$) graphs as a rectangular hyperbola: a curve in two separate pieces, called branches. It is undefined at $x = 0$ (you cannot divide by zero), so there is a gap there. A short table makes the shape clear: at $x = \tfrac12, 1, 2, 4$ the values are $y = 2, 1, \tfrac12, \tfrac14$, a curve falling steeply then flattening; the negative inputs give the matching negative outputs $y = -2, -1, -\tfrac12, -\tfrac14$. So one branch sits in the first quadrant (both coordinates positive) and one in the third (both negative); the function is odd, with point symmetry about the origin.

The defining feature is its two asymptotes, lines the curve approaches without ever meeting. As $x$ grows large the value $\dfrac{1}{x}$ shrinks toward $0$, so the curve hugs the $x$-axis: $y = 0$ is the horizontal asymptote. As $x$ approaches $0$ the value $\dfrac{1}{x}$ blows up in size, so the curve shoots up (or down) alongside the $y$-axis: $x = 0$ is the vertical asymptote. The curve gets arbitrarily close to each axis but never touches it, which is exactly what an asymptote means.

This is the graph of inverse variation: whenever one quantity is a fixed amount divided by another ($y = \dfrac{k}{x}$), the relationship is a rectangular hyperbola. A larger constant $k$ pushes the branches further from the origin (the curve passes through $(1, k)$), but the axes stay the asymptotes. Real examples are everywhere: at a fixed distance, travel time against speed; at a fixed area, the length of a rectangle against its width; at a fixed budget, quantity against unit price.

How exam questions ask about these graphs

The wording points to the shape and the method:

• "Sketch $y = x^n$" or "compare the graphs of $y = x^2$ and $y = x^3$." State even versus odd: a $y$-axis-symmetric U-shape for even powers, an origin-symmetric rising curve for odd, both through $(0, 0)$ and $(1, 1)$. Note the higher power is flatter near the origin.
• "Sketch the cubic / polynomial $y = (\ldots)(\ldots)(\ldots)$." Read the zeros from the factors, find the $y$-intercept, build a sign table with a test value per interval, set the end behaviour from the leading term, then join. Mark intercepts only.
• "Where does the curve lie above / below the $x$-axis?" Answer straight from the sign table, giving the intervals in $a < x < b$ form (for instance $-3 < x < 0$).
• "What happens at $x = \alpha$?" for a squared factor. The curve touches the $x$-axis and turns back there (the $x$-axis is a tangent), it does not cross.
• "Sketch / state the radius of the circle $x^2 + y^2 = \ldots$." Radius is the square root of the constant; centre is the origin; it cuts the axes at $\pm r$.
• "Show that the point $(a, b)$ lies on the circle." Substitute and show $a^2 + b^2$ equals $r^2$.
• "What does $y = \sqrt{r^2 - x^2}$ represent, and what are its domain and range?" The upper semicircle; domain $-r \le x \le r$, range $0 \le y \le r$.
• "Sketch $y = \dfrac{1}{x}$ (or $\dfrac{k}{x}$) and state its asymptotes." Two branches in the first and third quadrants; horizontal asymptote $y = 0$, vertical asymptote $x = 0$; the function is undefined at $x = 0$.
• "A quantity varies inversely as ... / find the constant of variation." Recognise $y = \dfrac{k}{x}$, find $k$ from a given pair, and describe the graph as a rectangular hyperbola.