Plot and draw graphs of linear functions y = mx + c, and interpret the gradient and y-intercept

What this dot point is asking

SEAB wants you to plot and draw the graph of a linear function written as $y = mx + c$, and to interpret what the gradient $m$ and the $y$-intercept $c$ mean. A linear function always gives a straight line, and reading that line is a skill used in travel graphs, coordinate geometry and statistics.

The answer

The equation of a straight line

Every straight line (except a vertical one) can be written as:

where $m$ is the gradient (steepness) and $c$ is the $y$-intercept (where the line crosses the $y$-axis). For $y = 3x + 2$, the gradient is $3$ and the line crosses the $y$-axis at $(0, 2)$.

Plotting a line from its equation

To draw the graph, build a small table of values:

1. Choose a few $x$ values, such as $0$, $1$ and $2$.
2. Work out $y$ for each using the equation.
3. Plot the points and join them with a straight line using a ruler.

For $y = 2x + 1$: when $x = 0$, $y = 1$; when $x = 1$, $y = 3$; when $x = 2$, $y = 5$. Three points that line up confirm there is no arithmetic slip.

Gradient

The gradient measures how steep the line is - how much $y$ changes for each unit increase in $x$:

A positive gradient rises from left to right; a negative gradient falls. A larger number means a steeper line. A gradient of $0$ is a horizontal line.

The y-intercept

The $y$-intercept $c$ is the value of $y$ where the line meets the $y$-axis, that is when $x = 0$. In a real context it often means a starting value, such as a fixed fee before any usage is added.

Reading values from a graph

Once a line is drawn, you can read off a $y$ value for any $x$ (or the reverse) by going up from the axis to the line and across. This is how travel and conversion graphs are used.

Examples in context

Example 1. A taxi fare. A taxi charges a \3$flag-down plus$\$1$ per kilometre, so the fare is $y = x + 3$ where $x$ is the distance. The $y$-intercept $3$ is the fixed starting charge and the gradient $1$ is the cost per kilometre. Reading the graph at $x = 5$ gives a fare of \8$.

Example 2. Comparing two lines. The lines $y = 2x$ and $y = 2x + 3$ have the same gradient, so they are parallel and never meet. The second is simply the first shifted up by $3$. Recognising equal gradients as parallel lines is a quick way to compare straight-line graphs.

Try this

• Cue. State the gradient and $y$-intercept of $y = 5x - 2$. The gradient is $5$ and the intercept is $-2$.
• Cue. Find the gradient of the line through $(2, 1)$ and $(6, 9)$. Compute $\dfrac{9 - 1}{6 - 2} = \dfrac{8}{4} = 2$.
• Cue. Find $y$ when $x = 3$ on the line $y = -2x + 10$. Substitute: $y = -6 + 10 = 4$.