2.05 Using linear functions

Finding the equation of a line

We are often asked to find the equation of a line given certain information about the graph or the context. Depending on the information given a particular form of a line might be more convenient. Here are some useful forms:

Let's look at examples where the different forms may be convenient. 

 

Gradient-intercept form

To use the gradient-intercept rule, we need to be able to find the gradient and the $y$y-intercept of a line. Then substitute these values into the form:

$y=mx+c$y=mx+c

where $m$m is the gradient and $c$c is the $y$y-intercept

If you are not given the gradient or $y$y-intercept directly, such as being given the gradient and a different point, you can still use this method by substituting the gradient and known point to find the unknown $y$y-intercept. Alternatively, to achieve this in one step we can use the point-gradient formula. 

Practice question

QUESTION 1

 

We may be given this information in context, remember the gradient represents the rate of change, for each increase in the $x$x-value by $1$1 we will see a change in the $y$y-value by $m$m. The $y$y-intercept is the value of $y$y when $x$x is zero, in context this is often the initial value. 

Practice question

QUESTION 2

 

Point-gradient formula

If we don't know the $y$y-intercept of the line but we do know a point ($x_1$x1​ , $y_1$y1​) on the line, we can use the point-gradient formula, which is:

$y-y_1=m\left(x-x_1\right)$y−y1​=m(x−x1​)

Practice question

Question 3

Question 4

 

General form

Another way of writing the equation of a straight line is called general form. A straight line is in general form when it is written with all terms on one side of the equation and zero on the other, with integer coefficients. In particular, the coefficient of $x$x should be positive (the sign can be easily changed by multiplying the whole equation by $-1$−1).

That is, a straight line is in general form when it is of the form

$ax+by+c=0$ax+by+c=0

for integers $a$a, $b$b and $c$c with $a>0$a>0. We can be asked to write our final answer in this form or it may be convenient to use this form to find the equation of a line parallel to another line given in this form.

From a graph or table

From a graph or table we will look for two easily identifiable points and use one of the methods above to find the equation. If it is in the table or clear on a graph one convenient point would be the $y$y-intercept. Then the gradient can be found using $m=\frac{rise}{run}$m=riserun​.

Practice questions

Question 5

Question 6