4.6.4 Key Features of Linear Functions

Features of Linear Functions

The graph of a linear function forms a line that goes on forever in both directions. Since the ends go to infinity and negative infinity, linear functions do not have maximums or minimums, unless their domain is restricted to a specific part of the graph.

You can still pick out key features of linear functions.

The $x$-intercept, or zero, is where the line crosses the $x$-axis. This is where the $y$-coordinate is 0.

In this linear function, the $x$-intercept, or zero, is $x=2$ and written as the point $(2,0)$.

The $y$-intercept is where the line crosses the $y$-axis and the $x$-coordinate is 0.

In this linear function, the $y$-intercept is $y=4$ and written as the point $(0,4)$.

The slope of the line can be found as in previous units as the change in $y$ over the change in $x$.

$\mathrm{m}=\frac{\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}}{\mathrm{r}\mathrm{u}\mathrm{n}}=\frac{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{y}}{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{x}}$

The slope of this linear function is $m=\frac{4}{-2}=\frac{-4}{2}=-2$.

Try it

Try It: Features of Linear Functions

Find the $x$-intercept and $y$-intercept of the linear function graphed below. Write each answer as a point.

Here is how to find the intercepts of a linear function:

Since the line crosses the $x$-axis at about $(-2.75,0)$, that is the $x$-intercept.

Since the line crosses the $y$-axis at $(0,2)$, that is the $y$-intercept.