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{{ printedBook.courseTrack.name }} {{ printedBook.name }} If a function has a constant rate of change, it is linear. Graphically, a linear function is a straight line.

Using that line, it's possible to determine the rate of change by finding the horizontal change $(Δx)$ and the vertical change $(Δy)$ between any two given points on the line. Any function whose graph is not a straight line cannot be linear.The slope of a line passing through the points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is the ratio of the vertical change $(Δy)$ to the horizontal change $(Δx)$ between the points. The variable $m$ is most commonly used to represent slope.

$m=ΔxΔy $

The words *rise* and *run* are sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to $Δy$ and run corresponds to $Δx.$

This gives the following definition for the slope of a line.

$m=ΔxΔy =runrise $

The sign (positive or negative) of each distance corresponds to the direction of the movement between points. Moving to the right, the run is positive; moving to the left, the run is negative. Similarly, moving up yields a positive rise while moving down gives a negative rise.

The slope of a line can be found algebraically using the following rule.

$m=x_{2}−x_{1}y_{2}−y_{1} $

One way to express linear function rules is called slope-intercept form.

$y=mx+b$

Sometimes, $f(x)=mx+b$ is used. In either case, $m$ and $b$ describe the general characteristics of the line. $m$ indicates the slope, and $b$ indicates the $y$-intercept. The linear function graphed below can be expressed as $f(x)=2x+1,$ because it has a slope of $2$ and a $y$-intercept at $(0,1).$

Write the equation of the line that passes through the points $(3,1)$ and $(5,5).$

Show Solution

A line in slope-intercept form is given by the equation
$y=mx+b,$
where $m$ is the slope of the line and $b$ is the $y$-intercept. Since two points on the line are known we can use the slope formula to determine the slope.
The equation, $y=mx+b,$ can now be rewritten using the knowledge that the slope, $m,$ is $2.$
$y=2x+b$
The $y$-intercept can now be found by replacing $x$ and $y$ in the equation with one of the points, for example $(5,5),$ and solve the resulting equation.
Thus, the equation of the line passing through the two points is
$y=2x−5.$

$y=2x+b$

$5=2⋅5+b$

MultiplyMultiply

$5=10+b$

SubEqn$LHS−10=RHS−10$

$-5=b$

RearrangeEqnRearrange equation

$b=-5$

To write the equation of the graph of a line in slope-intercept form, $y=mx+b,$

the $y$-intercept, $b$, and the slope of the line, $m,$ must be found. The following method can be used. As an example, consider the line shown.

Find the $y$-intercept

Replace $b$ with the $y$-intercept

Find the slope

Next, the slope of the line must be determined. From a graph, the slope of a line can be expressed as $m=runrise ,$ where rise gives the vertical distance between two points, and run gives the horizontal distance. To find the slope, use any two points. Here, use the already marked $y$-intercept and the arbitratily chosen $(2,2).$

From the lines drawn, it can be seen that the $rise=6$ and the $run=2.$ Therefore, the slope is $m=26 =3.$

Replace $m$ with the slope

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