Graph linear functions by constructing a table of values and plotting the points, recognise a linear relationship, sketch horizontal and vertical lines, and graph a line using its x- and y-intercepts

What this dot point is asking

A linear relationship is one where, every time the independent variable goes up by the same step, the dependent variable changes by the same amount. Plot it and the points fall in a perfectly straight line, which is why these are called linear functions. NESA wants you to turn such a relationship into a graph and read information back off it. Three skills are bundled here, and a question can test any of them. The first is to build a table of values and plot the points to draw the line. The second is to recognise and sketch the two special cases, horizontal lines $y = k$ and vertical lines $x = k$. The third is to graph a line quickly from its $x$- and $y$-intercepts when you are given its equation.

The deeper idea worth carrying through all of it is that two points are enough to fix a straight line. A table of values gives you several points as a safety net: if one is out of line, you have made an arithmetic slip. But once you trust the relationship is linear you only ever need two well-chosen points, and the two easiest to find are almost always the intercepts. The marks come from a correctly labelled and scaled set of axes, accurately plotted points, a ruled straight line, and reading values off cleanly. Marks are lost in predictable ways: cramped or unlabelled axes, a freehand wavy line, plotting a point with the coordinates the wrong way around, and misreading a horizontal line for a vertical one.

The answer

Recognising a linear relationship

When a relationship is linear, equal steps in $x$ produce equal steps in $y$. Look at the table for $y = 2x - 1$:

Every time $x$ increases by $1$, $y$ increases by $2$. That constant step is the fingerprint of a straight line: it is what makes the points line up when you plot them. A relationship whose $y$-steps are not constant (for example $y = x^2$, where the gaps grow) is not linear and its graph is a curve, which is outside this dot point. In Mathematics Standard, the linear functions you graph come in two shapes: written as $y = \ldots$ in terms of $x$ (such as $y = 2x - 1$ or $y = 4n$), or written as an equation like $3x + 2y = 12$ that mixes $x$ and $y$ on one side. The method below handles both.

Graphing from a table of values

This is the bread-and-butter method, and it always works. There are three moves: build the table, draw and label the axes, then plot and rule.

• Build the table of values. Put the independent variable (usually $x$) in the top row and choose a small spread of values, commonly a few either side of $0$. Substitute each one into the equation to fill in the bottom row (the dependent variable, usually $y$). Three points is the minimum, but four or five is safer because a stray point reveals a mistake.
• Draw and label the axes. Put the independent variable on the horizontal axis and the dependent variable on the vertical axis. Choose a scale that fits all your values and is evenly spaced, and label each axis with its variable and units. Good scaling is itself worth marks.
• Plot the points and rule the line. Plot each ordered pair $(x, y)$, going across first, then up or down. Then join them with a single straight line drawn with a ruler, extending it a little past the end points and adding arrowheads.

The four panels below build the graph of a real Australian context, the cost $C$ dollars of buying $n$ punnets of strawberries at a market stall charging $4$ dollars each, so $C = 4n$. The table of values is:

Stage 1, draw and label the axes. Put the independent variable $n$ (the number of punnets) on the horizontal axis and the dependent variable $C$ (the cost in dollars) on the vertical axis. Scale the cost axis in steps of $4$ so all the values from the table fit, and label each axis. An empty, well-scaled plane is the foundation; getting the scale right now saves redrawing later.

Stage 2, plot the points from the table. Take each pair from the table and plot it, reading the punnet value across the bottom and the cost value up the side: $(0, 0)$, $(1, 4)$, $(2, 8)$, $(3, 12)$, $(4, 16)$ and $(5, 20)$. Plot across first, then up. Already the dots look as if they line up, which is the visual check that the relationship is linear.

Stage 3, rule a straight line through the points. Lay a ruler along the points and draw one straight line through all of them, extending it a little past the ends. If a point sits off the line, recheck that row of the table; here all six points lie exactly on the line, confirming $C = 4n$ is linear.

Stage 4, read values off the finished graph. The graph now answers questions the table did not list. The line passes through the origin $(0, 0)$, its $y$-intercept, meaning zero punnets cost nothing. To find the cost of $2.5$ punnets-worth, read up from $2.5$ on the bottom axis to the line, then across to the cost axis, landing on $10$ dollars (and indeed $4 \times 2.5 = 10$).

Horizontal and vertical lines

Two special linear graphs are worth knowing on sight, because their equations look like they are missing a variable.

• A horizontal line has equation $y = k$ for some fixed number $k$. Every point on it has the same $y$-coordinate, $k$, no matter what $x$ is, so the line runs flat across the plane through $k$ on the $y$-axis. For example $y = 3$ passes through $(-3, 3)$, $(0, 3)$ and $(2, 3)$.
• A vertical line has equation $x = k$. Every point on it has the same $x$-coordinate, $k$, whatever $y$ is, so the line runs straight up and down through $k$ on the $x$-axis. For example $x = -2$ passes through $(-2, -1)$, $(-2, 0)$ and $(-2, 2)$.

The way to keep them straight is to ask which coordinate is pinned. In $y = 3$ the $y$-value is pinned at $3$ and $x$ roams free, so the line is the set of all points at height $3$: a horizontal line. In $x = -2$ the $x$-value is pinned and $y$ roams free, so the line is vertical. The pair below shows both on one plane.

Graphing a line from its intercepts

When you are given an equation like $3x + 2y = 12$, building a full table is slower than you need. Because two points fix a line, the fastest sketch comes from the two intercepts, the points where the line crosses the axes.

• The $x$-intercept is where the line crosses the $x$-axis. On the $x$-axis, $y = 0$, so set $y = 0$ and solve for $x$.
• The $y$-intercept is where the line crosses the $y$-axis. On the $y$-axis, $x = 0$, so set $x = 0$ and solve for $y$.

For $3x + 2y = 12$: setting $y = 0$ gives $3x = 12$, so $x = 4$ and the $x$-intercept is $(4, 0)$; setting $x = 0$ gives $2y = 12$, so $y = 6$ and the $y$-intercept is $(0, 6)$. Mark those two points and rule the line through them. The diagram below shows the result, with both intercepts marked.

The trick to remember which to zero is that you head to an axis by killing the other variable: to reach the $x$-axis you make $y$ zero, and to reach the $y$-axis you make $x$ zero. It feels back to front at first, which is exactly why it is a common trap.

How exam questions ask about graphing linear functions

The wording tells you which method to reach for:

• "Complete the table of values" or "draw the graph of $y = \ldots$ by first constructing a table of values" means substitute the listed $x$ values, plot the pairs, and rule the line.
• "Plot the following points and join them to form a straight line" gives you the table already; just plot accurately and rule.
• "Sketch the graph of $x = 4$" (or $y = -1$) is testing the special lines: decide horizontal or vertical by which coordinate is pinned, then draw it.
• "Find the $x$-intercept and the $y$-intercept", or "sketch the line $ax + by = c$", means use the intercept method: zero $y$ for the $x$-intercept, zero $x$ for the $y$-intercept.
• "Use your graph to find ..." is a read-off: go up (or across) from the given value to the line, then across (or down) to the other axis, and state the value with units.
• "Show that the point $(\ldots)$ lies on the line" means substitute the coordinates into the equation and check both sides are equal.

Why two points (the intercepts) are enough

It is worth seeing why the intercept shortcut is valid, because it is the reasoning behind nearly every quick sketch. A straight line has no bends, so once you know any two distinct points on it there is exactly one line you can rule through them: two points determine a line. A table of values gives you more than two points, but only as insurance against an arithmetic slip; the line itself is already pinned down by any two of them. The intercepts are simply the two easiest points to compute, because setting a variable to zero removes a whole term from the equation.

The method does have one edge case. A line through the origin, like $C = 4n$ or $d = 80t$, has both intercepts at the same point $(0, 0)$, so the two intercepts coincide. Here you need a second point elsewhere to fix the direction, found by substituting any convenient value, say $n = 5$. Lines of the form $y = k$ and $x = k$ are the other edge case. A horizontal line $y = k$ (with $k \neq 0$) never crosses the $x$-axis, and a vertical line $x = k$ (with $k \neq 0$) never crosses the $y$-axis, so the intercept method does not apply and you read them straight off as flat or upright.