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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.

y=mx+b

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

$m=2=12 andb=-3.$

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.

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.

The graph above shows the function f(x)=x+3. It can be seen that, at 6AM, 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 $f(x)=-32 x+4.$ 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,
*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.

$x=6andy=4.$

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, $Decreasing Interval:-∞<x<+∞$

Since f always decreases, we can say f is 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.

$Asx→-∞,Asx→+∞, f(x)→+∞f(x)→--∞ $

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
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.

$function rule:y=2.5x−1.5equation:6=2.5x−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=2.5x−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. Since a true statement is made, we can confirm that the solution to the equation is x=3. {{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

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