C3.5 Equations of linear graphs

Learning objectives

C3.5.1Interpret and obtain the equation of a straight-line graph in the form• Interpret and obtain the equation of a straight-line graph in the form y = mx + c. Questions may: • use and request lines in the forms y = mx + c x = k • involve finding the equation when the graph is given • ask for the gradient or y-intercept of a graph from an equation, e.g. find the gradient and y-intercept of the graph with the equation y = 6x + 3. Candidates are expected to give equations of a line in a fully simplified form.

In $y=mx+c$ , $m$ is the line’s gradient and $c$ is its $y$ -intercept value. These two numbers completely determine a non-vertical straight line.

Feature	How to read it
gradient	coefficient $m$ of $x$
$y$ -intercept	$c$ , giving point $(0,c)$
horizontal line	$y=k$
vertical line	$x=k$

For $y=6x+3$ , the gradient is $6$ and the line crosses the $y$ -axis at $(0,3)$ . Rewrite an equation into fully simplified $y=mx+c$ form before reading these values.

To obtain an equation from a graph: read the axis scales; choose two exact points and calculate $m=\dfrac{ ext{vertical change}}{ ext{horizontal change}}$ ; read the value $c$ where the line crosses the $y$ -axis; substitute both into $y=mx+c$ .

$m=\frac{6-1}{0-1}=-5,\qquad c=6\quad\Longrightarrow\quad y=-5x+6$

Check the equation against another point on the graph. For example, $(1,1)$ satisfies $y=-5x+6$ because $-5(1)+6=1$ . In a context, use the variables named in the question, such as $c=2d+5$ .

Do not write a vertical line as $y=mx+c$ : its equation is $x=k$ . Always include $y=$ (or the named dependent variable) and give the final equation in fully simplified form.

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—C3.6 Parallel linesRead