6.3Linear Functions: The Equation y = mx + c

Recall: Chapter 5.1 (Linear Equations), the balancing principle used to rearrange an equation into

y = mx + c

.→

A linear function has the form $y = mx + c$ : since the highest power of $x$ is 1, its graph is always a straight line. The constant $m$ sets the steepness; the constant $c$ sets where the line crosses the $y$ -axis.

Standard form of a linear equation

\displaystyle y = mx + c

$m$ is the gradient: positive means the line rises, negative means it falls, zero means it is horizontal.
$c$ is the $y$ -intercept: the value of $y$ when $x = 0$ .
The further $m$ is from zero, in either direction, the steeper the line.

The line

y = 2x + 1

has gradient

m = 2

and

y

-intercept

c = 1

, crossing the

y

-axis at

(0, 1)

Reading gradient and $y$ -intercept

State the gradient and $\displaystyle y$ -intercept of the line $\displaystyle y = -4x + 9$ .

Show solution▾

Comparing

y = -4x + 9

with

y = mx + c

Gradient

m = -4

y

-intercept

c = 9

Since

m < 0

, the line falls from left to right.

Now you try

State the gradient of $y = -3x$ .

Show solution▾

y = -3x

This is

y = mx

where

c = 0

Gradient

m = -3

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Rearranging to find gradient and $y$ -intercept

Find the gradient and $\displaystyle y$ -intercept of the line $\displaystyle 3y + 12x = 15$ .

Show solution▾

\displaystyle 3y + 12x

\displaystyle = 15

\displaystyle 3y

\displaystyle = -12x + 15 \quad \scriptsize\textit{subtract 12x from both sides}

\displaystyle y

\displaystyle = -4x + 5 \quad \scriptsize\textit{divide both sides by 3}

Gradient

m = -4

y

-intercept

c = 5

Now you try

State the $y$ -intercept of $y = -x - 2$ .

Show solution▾

y = -x - 2

Comparing with

y = mx + c

y

-intercept

c = -2

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6.3Linear Functions: The Equation y = mx + c

Reading gradient and yyy-intercept

Rearranging to find gradient and yyy-intercept

Practise this

Reading gradient and $y$ -intercept

Rearranging to find gradient and $y$ -intercept