Identifying key features
The graph of a linear relationship will create a line. All linear functions can be written in either of these two common forms:
| Gradient intercept form | General form |
|---|---|
| $y=mx+c$y=mx+c | $ax+by+c=0$ax+by+c=0 |
Sketching a linear relationship on a plane, the straight line formed will be one of the following types:
- An increasing graph means that as $x$x values increase, the $y$y values increase.
- A decreasing graph means that as $x$x values increase, the $y$y values decrease.
- A horizontal graph means that as $x$x values change the $y$y values remain the same.
- In a vertical graph, the $x$x value is constant.
Regardless of these different shapes, all linear functions have some common characteristics.
Intercepts
Linear graphs all have at least one intercept. Linear functions might have
- an $x$x intercept only (in the case of a vertical line)
- a $y$y intercept only (in the case of horizontal lines)
- or $2$2 intercepts, both an $x$x and a $y$y (in the case of increasing or decreasing functions)
The $x$x intercept occurs at the point where $y=0$y=0.
The $y$y intercept occurs at the point where $x=0$x=0.
Gradient
The gradient (slope) of a line is a measure of how steep the line is. For a linear function, the gradient is constant. That is, as the $x$x-value increases by a constant amount, the $y$y-value also increases by a constant amount. The gradient can be calculated from any two points $\left(x_1,y_1\right)$(x1,y1), $\left(x_2,y_2\right)$(x2,y2) on a line:
| $m$m | $=$= | $\frac{rise}{run}$riserun |
| $=$= | $\frac{y_2-y_1}{x_2-x_1}$y2−y1x2−x1 |
The gradient is often represented by the letter $m$m and has the following properties:
- 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
- For vertical lines, $m$m is undefined
- The slope can be read directly, as the coefficient of $x$x, from the gradient intercept form ($y=mx+c$y=mx+c)
- By rearranging a linear equation in general form ($ax+by+c=0$ax+by+c=0 ), the gradient will be $m=-\frac{a}{b}$m=−ab
- The gradient states that as the $x$x-value increases by $1$1, the $y$y-value changes by $m$m
- The larger the value of $\left|m\right|$|m|, the steeper the line. Here, | | refers to the absolute value or modulus of $m$m (the positive value of $m$m).
Practice questions
Question 1
A line has the following equation: $y=6\left(3x-2\right)$y=6(3x−2)
Rewrite $y=6\left(3x-2\right)$y=6(3x−2) in the form $y=mx+c$y=mx+c.
State the gradient and $y$y-value of the $y$y-intercept of the equation.
Gradient $\editable{}$ Value at $y$y-intercept $\editable{}$
Question 2
Would the following table of values represent a linear graph?
$x$x $3$3 $6$6 $9$9 $12$12 $15$15 $y$y $-7$−7 $-14$−14 $-21$−21 $-28$−28 $-35$−35 Yes
ANo
B$x$x $7$7 $21$21 $35$35 $49$49 $63$63 $y$y $\frac{15}{2}$152 $15$15 $\frac{45}{2}$452 $45$45 $\frac{135}{2}$1352 Yes
ANo
B
Question 3
What is the gradient of the line going through A $\left(-1,1\right)$(−1,1) and B $\left(5,2\right)$(5,2)?