Here is a quick recap of what we know about straight lines on the coordinate plane:
Any straight line on the coordinate plane is defined entirely by its slope and its $y$y-intercept. These two features are all we need to write an equation for the line.
The variable $a$a is used to represent the slope, and the variable $b$b is used for the $y$y-intercept.
We can represent the equation of any straight line, except vertical lines, using what is known as the slope-intercept form of a straight line.
$y=ax+b$y=ax+b,
also known as $y=mx+b$y=mx+b or $y=mx+c$y=mx+c.
To represent a particular line on the coordinate plane, with it's own unique slope and $y$y-intercept, we simply replace $a$a and $b$b with their corresponding values.
We can use the applet below to see the effect of varying $a$a and $b$b on both the line and its equation.
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As can be seen from the applet, we can make the following statements about the slope.
The value of the slope, $a$a, relates to the line as follows:
In algebra, any number written immediately in front of a variable, is called a coefficient. For example, in the term $3x$3x, the coefficient of $x$x is $3$3. Any number by itself is known as a constant term.
In the slope-intercept form of a line, $y=ax+b$y=ax+b, the slope, $a$a, is the coefficient of $x$x, and the $y$y-intercept, $b$b, is a constant term.
If we know the slope, $a$a, and $y$y-intercept, $b$b, of a line, we can substitute these values into $y=ax+b$y=ax+b to write the equation of the line.
Write the equation of a line that has a slope of $\frac{3}{4}$34 and a $y$y-intercept of $-2$−2.
Solution
Substitute $a=\frac{3}{4}$a=34 and $b=-2$b=−2 into $y=ax+b$y=ax+b
$y$y | $=$= | $ax+b$ax+b |
$y$y | $=$= | $\frac{3}{4}x+\left(-2\right)$34x+(−2) |
$y$y | $=$= | $\frac{3}{4}x-2$34x−2 |
Write down the equation of a line which has a slope of $-3$−3 and crosses the $y$y-axis at $-9$−9.
Give your answer in slope-intercept form.
State the slope and $y$y-intercept of the equation $y=-8x-8$y=−8x−8.
Slope | $\editable{}$ |
$y$y-intercept | $\editable{}$ |
To graph a line from an equation, we use the slope, $a$a, and the $y$y-intercept, $b$b.
We begin by locating the $y$y-intercept as a point on the $y$y-axis.
From this point, we can use the slope to draw the correct slope of the line, as outlined in the three examples below:
Graph the line that has a slope of $4$4 and a $y$y-intercept of $-1$−1.
Solution
By comparing the equation of the line with $y=ax+b$y=ax+b, we see that the slope is $4$4 and the y-intercept is $-1$−1. The slope ($4=\frac{4}{1}$4=41) tells us that for a 'run' of $1$1, we have a 'rise' of $4$4.
We can now create the graph on a coordinate plane, in a series of steps:
Graph the line with equation $y=-\frac{2}{3}x$y=−23x.
Solution
By comparing the equation of the line with $y=ax+b$y=ax+b, we see that the slope is $-\frac{2}{3}$−23 and the $y$y-intercept is $0$0, meaning the line passes through the origin. The slope tells us that for a 'run' of $3$3, we have a 'rise' of $-2$−2.
We can now create the graph on a coordinate plane, in a series of steps:
Sketch a graph of the linear equation $y=4x+3$y=4x+3.
A horizontal line has a slope of zero ($a=0$a=0), so the equation of the line becomes:
$y=b$y=b
where $b$b is the $y$y-intercept of the line.
Here are two examples of horizontal lines:
A vertical line has an undefined slope, so we can't use the slope-intercept form to write its equation. Instead we define a vertical line as having the equation:
$x=c$x=c
where $c$c is the $x$x-intercept of the line.
Here are two examples of vertical lines:
Write down the equation of the graphed line.