Gradients, Intercepts and Sketching Linear Graphs

What a straight-line equation describes

A coordinate pair $(x,y)$ lies on a graph when its values satisfy the graph's equation. For example, $(2,7)$ lies on $y=3x+1$ because $3(2)+1=7$. A straight-line equation gives a constant relationship between $x$ and $y$: equal changes in $x$ always produce equal changes in $y$.

A non-vertical straight line can be written in the form

$y=mx+c$

Here, $m$ is the gradient and $c$ is the $y$-intercept. These two numbers determine the line: $m$ controls its direction and steepness, while fixes its vertical position.

Understanding the gradient

Gradient compares a vertical change with the corresponding horizontal change:

$m=\frac{\text{change in }y}{\text{change in }x}=\frac{\text{rise}}{\text{run}}$

Suppose moving 4 units to the right along a line takes you 6 units upwards. Its gradient is $\frac{6}{4}=\frac32$. This means that $$ increases by for every increase of 1 in . Equivalently, the line rises 3 units for every 2 units it moves right.

The sign and size of $m$ describe the graph:

• $m>0$: the line rises from left to right;
• $m<0$: the line falls from left to right;
• $m=0$: the line is horizontal;
• a larger value of $|m|$ gives a steeper line.

For instance, $y=-2x+5$ has gradient $-2$. Increasing $x$ by 1 decreases $y$ by 2. Its line therefore falls from left to right. On a straight line the gradient is constant, so any suitable pair of exact points gives the same result.

Finding a gradient from a graph

Choose two clear points on the line and form a right-angled gradient triangle. Read the actual changes from the axis scales, not just the number of grid squares. Then calculate vertical change divided by horizontal change.

If the chosen points are $(1,5)$ and $(4,-1)$, moving from the first to the second gives

$\text{change in }x=4-1=3$

and

$\text{change in }y=-1-5=-6.$

Therefore $m=\frac{-6}{3}=-2$. Both differences must describe the same direction of travel. Reversing the direction changes both signs, so the quotient remains $-2$.

Understanding intercepts

An intercept is where a graph crosses an axis. At every point on the $y$-axis, $x=0$. Substituting $x=0$ into $y=mx+c$ gives , so the -intercept is always .

For $y=4x-3$, the gradient is $4$ and the $y$-intercept is $($. Notice the distinction between the intercept value, , and the coordinate of the intercept, .

The $x$-intercept is where $y=0$. For the same line,

$0=4x-3,$

so $x=\frac34$ and the $x$-intercept is $\left(\frac34,0\right)$. Although specifically represents the -intercept, finding both intercepts can provide two convenient points for a .

In the visual, the actual coordinate axes cross at $(0,0)$. The line $y=2x+1$ crosses the $y$-axis at $(0,1)$ and rises 2 units whenever increases by 1. The line crosses the -axis at and falls 1 unit for every unit moved right. This shows that the intercept and gradient describe different features of a line.

Reading $m$ and $c$ from an equation

You may read $m$ and $c$ directly only when the equation has been arranged as $y=mx+c$.

For example, in $y=5x-7$, $m=5$ and $c=-7$. If no constant is shown, it is zero: $y=\frac{3}{}$ means , so the graph passes through the origin.

Other equations must first be rearranged by making $y$ the subject. For example,

$2y=6x+8$

becomes

$y=3x+4$

after every term is divided by 2. Thus the gradient is 3 and the $y$-intercept is $(0,4)$.

Similarly,

$3x+y=7$

becomes

$y=-3x+7,$

so its gradient is $-3$, not $3$, and its $y$-intercept is $(0,7)$.

Finding an equation from a graph

To identify a line's equation from its graph:

1. Read the point where it crosses the $y$-axis; its $y$-coordinate is $c$.
2. Use two exact points to calculate $m=\frac{\text{change in }y}{\text{change in }x}$.

If a graph crosses the $y$-axis at $(0,3)$ and falls 2 units while moving 4 units right, then

$m=\frac{-2}{4}=-\frac12$

and its equation is

$y=-\frac12x+3.$

This equation predicts every point on the line, not just the two used to determine the gradient.

Sketching from $y=mx+c$

A sketch must show the correct intercept, direction and steepness. Begin by plotting $(0,c)$. Treat the gradient as a rise divided by a run and use it to locate at least one more point. Draw a straight line through the points and label it.

To sketch $y=\frac12x-2$:

• plot the $y$-intercept $(0,-2)$;
• write $\frac12$ as rise $$ over run ;

For $y=-\frac23x+4$, start at $(0,4)$. Moving 3 units right requires moving 2 units down, giving another point at . A negative gradient changes the direction of the vertical move.

A table of values is an alternative: select several $x$-values, substitute them into the equation and plot the resulting coordinate pairs. Two correct points determine a straight line, but a third point is a useful check. Intercepts can also be used: set $x=0$ for the $y$-intercept and $y=0$ for the $x$-intercept, then join the two points.

Lines outside the form $y=mx+c$

A horizontal line such as $y=4$ has gradient 0 and $y$-intercept $(0,4)$, so it is equivalent to .

A vertical line such as $x=4$ cannot be written as $y=mx+c$. Its horizontal change is zero, which would require division by zero in the gradient calculation, so its gradient is undefined. It should be sketched as a vertical line through 4 on the $x$-axis.

Core form

$y=mx+c$

• $m$: gradient
• $c$: - value, giving the point