So what you will have found is that the $m$m value affects the gradient.
If $m<0$m<0, the gradient is negative and the line is decreasing
if $m>0$m>0, the gradient is positive and the line is increasing
if $m=0$m=0 the gradient is $0$0 and the line is horizontal
Also, the larger the value of $m$m the steeper the line
Y-Intercept
We also found that the $b$b value affects the $y$y intercept.
If $b$b is positive then the line is vertically translated (moved) up.
If $b$b is negative then the line is vertically translated (moved) down.
Transformations of the Line
So from equations in this form, $y=mx+b$y=mx+b, we instantly have enough information to understand what this line looks like and to describe the transformations from the basic line $y=x$y=x.
Examples
By first identifying the gradient and $y$y intercept, describe the transformations of the following lines from the basic line $y=x$y=x.
Question 1
$y=3x$y=3x
gradient is $3$3
$y$y intercept is $0$0
Transformations of change: The line $y=x$y=x is made steeper due to a gradient of $3$3 and is not vertically translated (it has the same $y$y-intercept as $y=x$y=x).
Question 2
$y=-2x$y=−2x
gradient is $-2$−2
$y$y intercept is $0$0
Transformations of change: The line $y=x$y=x is made steeper due to a gradient of $2$2, is reflected on the $x$x-axis (due to a negative gradient), and is not vertically translated.
Question 3
$y=\frac{x}{2}-3$y=x2−3
gradient is $\frac{1}{2}$12
$y$y intercept is $-3$−3
Transformations of change: the line $y=x$y=x is made less steep due to a gradient of $\frac{1}{2}$12 and is vertically translated $3$3 units down (a $y$y-intercept of $-3$−3 compared to a $y$y-intercept of $0$0 in $y=x$y=x).
Question 4
$2y=-4x+10$2y=−4x+10
First we need to rewrite it in the gradient intercept form.
$y=-2x+5$y=−2x+5
gradient is $-2$−2
$y$y intercept is $5$5
Transformations of change: the line $y=x$y=x is made more steep due to a gradient of $2$2, and is reflected on the $x$x-axis (due to a negative gradient). It is vertically translated $5$5 units up.
Creating equations from information given about the line
To create an equation of the form $y=mx+b$y=mx+b, we need 2 pieces of information: if we know the gradient and the $y$y-intercept, we can instantly write down the equation.
Example
What is the equations of the line with the a gradient of $\frac{3}{4}$34 and a $y$y intercept of $-2$−2?
The equation of the line will be:
$y=mx+b$y=mx+b
$y=\frac{3}{4}x-2$y=34x−2
Here are some worked examples.
Question 1
It is easier to read the gradient and $y$y-intercept from a linear equation if you rearrange the equation into gradient-intercept form:
$y=mx+b$y=mx+b
What is the gradient of the line $y=\frac{3-2x}{8}$y=3−2x8?
Question 2
Given that the line $y=mx+c$y=mx+c has a gradient of $-2$−2 and passes through $\left(-6,-3\right)$(−6,−3):
Find $c$c, the value of the $y$y-intercept of the line.
Find the equation of the line in the form $y=mx+c$y=mx+c.
Outcomes
11.CG.SL.1
Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axes, point-slope form, slope-intercept form, two-point form, intercepts form and normal form. General equation of a line. Distance of a point from a line.