Graphing Linear Functions in Slope-Intercept Form

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.

Concept

Slope-Intercept Form

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

$y = m x + b$

Sometimes, $f (x) = m x + b$ 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 $f (x) = 2 x + 1,$ because it has a slope of $2$ and a $y$ -intercept at $(0, 1) .$

Method

Graphing a Linear Function in Slope-Intercept Form

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

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

Graph the function

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Exercise

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.

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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 $f (x) = \frac{1}{1} x + 3 \Rightarrow f (x) = x + 3$ 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.

Concept

Describing Key Features of Linear Functions

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.

Describe the key features of the linear function

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Exercise

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

Show Solution

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, $x = 6 and y = 4 .$ 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, $Decreasing Interval : - \infty < x < + \infty$ 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. $As x \to - \infty, As x \to + \infty, f (x) \to + \infty f (x) \to - - \infty$ Lastly, we'll label the features on the graph.

Method

Solving Linear Equations Graphically

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 $(- 1, 2), (0, 3), (1, 4), (2, 5) .$

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 to solve the equation

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Exercise

Use the graph of $y = 2.5 x - 1.5$ to solve the equation $6 = 2.5 x - 1.5 .$

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Solution

The given graph shows all of the $x$ - $y$ pairs that satisfy $y = 2.5 x - 1.5 .$ Let's begin by comparing the function rule with the equation. $function rule : y = 2.5 x - 1.5 equation : 6 = 2.5 x - 1.5$ 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 $6 = 2.5 x - 1.5$ means finding the $x$ -coordinate of the point on the graph of $y = 2.5 x - 1.5$ 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 = 2.5 x - 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 = 2.5 x - 1.5

Substitute

$x = 3$

6 = ? 2.5 \cdot 3 - 1.5

Multiply

6 = ? 7.5 - 1.5

SubTerm

Subtract term

6 = 6

Since a true statement is made, we can confirm that the solution to the equation is

x = 3 .

Graphing Linear Functions in Slope-Intercept Form

Concept

Slope-Intercept Form

Method

Graphing a Linear Function in Slope-Intercept Form

Example

Graph the function

Concept

Describing Key Features of Linear Functions

Example

Describe the key features of the linear function

Method

Solving Linear Equations Graphically

Example

Use the graph to solve the equation