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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.
Rule

Slope-Intercept Form

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

y=mx+by=mx+b

Sometimes, f(x)=mx+bf(x)=mx+b is used. In either case, mm and bb describe the general characteristics of the line. mm indicates the slope, and bb indicates the yy-intercept. The linear function graphed below can be expressed as f(x)=2x+1, f(x)=2x+1, because it has a slope of 22 and a yy-intercept at (0,1).(0,1).

Method

Graphing a Linear Function in Slope-Intercept Form

To graph a line in slope-intercept form, the slope, m,m, and the yy-intercept, b,b, are both needed. Consider the linear function y=2x3.y=2x-3. Since the rule is written as y=mx+b,y=mx+b, it can be seen that m=2=21andb=-3. m=2 = \dfrac{2}{1} \quad \text{and} \quad b=\text{-} 3. To graph the line, plot the yy-intercept, then use the slope to find another point on the line. Specifically, from (0,-3),(0,\text{-} 3), move up 22 units and right 11 unit.

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

Exercise

In Clear Lake, Iowa, during a particular evening, there is a 33-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 66 AM.

Solution

To begin, we can define the quantities that xx and yy represent.

  • Let xx represent the number of hours it's been snowing.
  • Let yy represent the number of inches of snow on the ground.

It's been given that there is a 33-inch layer of snow on the ground before it begins to snow. Thus, when x=0,y=3.x=0, y=3. In other words, the yy-intercept is b=3.b=3. It is also given that the amount of snow on the ground increases by 11 inch every 11 hour. Thus, the slope of the line is 11.\frac{1}{1}. We can write the rule for this function as f(x)=11x+3f(x)=x+3 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),(0,3), then move up 11 unit and right 11 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.f(x)=x+3. It can be seen that, at 66AM, there is a total of 99 inches of snow on the ground.

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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.
Exercise

The graph shows the function f(x)=-23x+4.f(x)=\text{-} \frac{2}{3}x+4. Describe the intercepts, increasing/decreasing intervals, and end behavior. Then, show the features on the graph.

Solution

To begin, we'll describe each of the features. A graph's xx- and yy-intercepts are the points where the graph intersects the xx- and yy-axes, respectively. It can be seen that ff intersects the xx-axis at (6,0)(6,0) and the yy-axis at (0,4).(0,4). Thus, x=6andy=4. x=6 \quad \text{and} \quad y=4. Since ff 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 xx increases, f(x)f(x) decreases infinitely. Thus, Decreasing Interval:-<x<+ \textbf{Decreasing Interval:} \quad \text{-} \infty < x < + \infty Since ff always decreases, we can say ff is negative. Notice that this corresponds with ff'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 ff can be written as follows. As x-, f(x)+As x+, f(x)--\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.

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Method

Solving Linear Equations Graphically

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

All of the xx-yy 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.
Exercise

Use the graph of y=2.5x1.5y=2.5x-1.5 to solve the equation 6=2.5x1.5.6=2.5x-1.5.

Solution

The given graph shows all of the xx-yy pairs that satisfy y=2.5x1.5.y=2.5x-1.5. Let's begin by comparing the function rule with the equation. function rule:y=2.5x1.5equation:6=2.5x1.5\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. y=6. Thus, solving the equation 6=2.5x1.56=2.5x-1.5 means finding the xx-coordinate of the point on the graph of y=2.5x1.5y=2.5x-1.5 whose yy-coordinate is 6.6. We'll find this point by first locating the point on the graph where y=6.y=6.

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

Since the desired point on the graph is (3,6),(3,6), the solution to 6=2.5x1.56=2.5x-1.5 is x=3.x=3. We can verify this value by substituting it into the equation to see if a true statement is made.
6=2.5x1.56=2.5x-1.5
6=?2.531.56 \stackrel{?}{=} 2.5 \cdot {\color{#0000FF}{3}}-1.5
6=?7.51.56 \stackrel{?}{=}7.5 -1.5
6=66 =6
Since a true statement is made, we can confirm that the solution to the equation is x=3.x=3.
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