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There are different ways to graph a linear function. Sometimes, the way the rule of the function is written, can dictate the simplest way to graph it. Below, the graphs of linear functions given in *slope-intercept form* will be explored.

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 $(0,1).$

To graph a line in slope-intercept form, the slope, $m,$ and the $y$-intercept, $b,$ are both needed. Consider the linear function $y=2x-3.$ Since the rule is written as $y=mx+b,$ it can be seen that $m=2 = \dfrac{2}{1} \quad \text{and} \quad b=\text{-} 3.$ To graph the line, plot the $y$-intercept, then use the slope to find another point on the line. Specifically, from $(0,\text{-} 3),$ move up $2$ units and right $1$ unit.

Next, draw a line through both points to create the graph of the linear function.

In Clear Lake, Iowa, during a particular evening, there is a $3$-inch layer of snow on the ground. At midnight, it begins to snow. Each hour, one inch of snow falls. Graph a function that shows the amount of snow on the ground from midnight to $6$ AM.

Show Solution

To begin, we can define the quantities that $x$ and $y$ represent.

- Let $x$ represent the number of hours it's been snowing.
- Let $y$ represent the number of inches of snow on the ground.

It's been given that there is a $3$-inch layer of snow on the ground before it begins to snow. Thus, when $x=0, y=3.$ In other words, the $y$-intercept is $b=3.$ It is also given that the amount of snow on the ground increases by $1$ inch every $1$ hour. Thus, the slope of the line is $\frac{1}{1}.$ We can write the rule for this function as $f(x)=\dfrac{1}{1}x+3 \quad \Rightarrow f(x)=x+3$ To graph the function, we can plot a point at $(0,3),$ then move up $1$ unit and right $1$ unit to find another point. The line that connects these points is the graph of the function.

The graph above shows the function $f(x)=x+3.$ It can be seen that, at $6$AM, there is a total of $9$ inches of snow on the ground.

Key features are certain characteristics of the graphs of functions that are noteworthy. For linear functions, a graph's intercepts, increasing/decreasing intervals, and end behavior are important.

The graph shows the function $f(x)=\text{-} \frac{2}{3}x+4.$ Describe the intercepts, increasing/decreasing intervals, and end behavior. Then, show the features on the graph.

Show Solution

To begin, we'll describe each of the features. A graph's $x$- and $y$-intercepts are the points where the graph intersects the $x$- and $y$-axes, respectively. It can be seen that $f$ intersects the $x$-axis at $(6,0)$ and the $y$-axis at $(0,4).$ Thus,
$x=6 \quad \text{and} \quad y=4.$
Since $f$ is a linear function, it has a constant rate of change. This means the line will always be increasing or always decreasing. Looking from left to right on the graph, it can be seen that, as $x$ increases, $f(x)$ decreases infinitely. Thus, $\textbf{Decreasing Interval:} \quad \text{-} \infty < x < + \infty$
Since $f$ always decreases, we can say $f$ is *negative.* Notice that this corresponds with $f$'s negative slope.

Looking at the graph, we can see that the left end extends upward and the right end extends downward. Thus, the end behavior of $f$ can be written as follows.
$\begin{aligned}
\text{As}\ x \rightarrow \text{-} \infty , && \ f(x) \rightarrow +\infty \\
\text{As}\ x \rightarrow +\infty , && \ f(x) \rightarrow \phantom{\text{-}} \text{-} \infty
\end{aligned}$
Lastly, we'll label the features on the graph.

The function $y=f(x)$ is comprised of all the $(x,y)$ points that satisfy its function rule. For example, some of the points that lie on the line $y=x+3$ are $(\text{-} 1,2), \quad (0,3), \quad (1,4), \quad (2,5).$

All of the $x$-$y$ pairs that satisfy an equation are the points on the corresponding graph. Thus, the solution(s) to an equation can be found using a graph.Use the graph of $y=2.5x-1.5$ to solve the equation $6=2.5x-1.5.$

Show Solution

The given graph shows all of the $x$-$y$ pairs that satisfy $y=2.5x-1.5.$ Let's begin by comparing the function rule with the equation. $\begin{aligned} \textbf{function rule} \quad : \quad y=2.5x-1.5\\ \textbf{equation} \quad : \quad 6=2.5x-1.5\\ \end{aligned}$ Since the right-hand side of both the function rule and the equation are the same, it follows then that $y=6.$ Thus, solving the equation $6=2.5x-1.5$ means finding the $x$-coordinate of the point on the graph of $y=2.5x-1.5$ whose $y$-coordinate is $6.$ We'll find this point by first locating the point on the graph where $y=6.$

We can move down from this point on the graph to find the $x$-coordinate.

Since the desired point on the graph is $(3,6),$ the solution to $6=2.5x-1.5$ is $x=3.$ We can verify this value by substituting it into the equation to see if a true statement is made.$6=2.5x-1.5$

Substitute$x={\color{#0000FF}{3}}$

$6 \stackrel{?}{=} 2.5 \cdot {\color{#0000FF}{3}}-1.5$

MultiplyMultiply

$6 \stackrel{?}{=}7.5 -1.5$

SubTermSubtract term

$6 =6$

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