2.5 Graphs of Linear Functions
When we are working with a new function, it is useful to know as much as we can about the function: its graph, where the function is zero, and any other special behaviors of the function.
When graphing a linear function, there are two basic ways to graph it:
- By plotting points (at least 2) and drawing a line through the points
- Using the initial value (output when ) and rate of change (slope)
Example Graphing a Line with Points
Graph by plotting points.
In general, we evaluate the function at two or more inputs to find at least two points on the graph. Usually it is best to pick input values that will “work nicely” in the equation.
In this equation, multiples of 3 will work nicely due to the \dfrac{2}{3} in the equation, and of course using to get the vertical intercept. Evaluating at and :
These evaluations tell us that the points (0,5), (3,3), and (6,1) lie on the graph of the line. Plotting these points and drawing a line through them gives us the graph.
Graphing a line using the initial value and rate of change
When using the initial value and rate of change to graph, we need to consider the graphical interpretation of these values. Remember the initial value of the function is the output when the input is zero, so in the equation , the graph includes the point . On the graph, this is the vertical intercept – the point where the graph crosses the vertical axis.
For the rate of change, it is helpful to remember that we calculated this value as:
From a graph of a line, this tells us that if we divide the vertical difference, or rise, of the function outputs by the horizontal difference, or run, of the inputs, we will obtain the rate of change, also called slope of the line.
Notice that this ratio is the same regardless of which two points we use.
Graphical Interpretation of a Linear Equation
Graphically, in the equation
is the vertical intercept of the grpah and tells us we can start at
is the slope of the line and tells us how far to rise & run to get to the next point.
Once we have at least 2 points, we can extend the graph of the line to the left and right.
Example Graphing a Line using Slope and Intercept
Graph using the vertical intercept and slope.
The vertical intercept of the function is (0, 5), giving us a point on the graph of the line.
The slope is . This tells us that for every 3 units the graph “runs” in the horizontal, the vertical “rise” decreases by 2 units. In graphing, we can use this by first plotting our vertical intercept on the graph, then using the slope to find a second point. From the initial value (0, 5) the slope tells us that if we move to the right 3, we will move down 2, moving us to the point (3, 3). We can continue this again to find a third point at (6, 1). Finally, extend the line to the left and right, containing these points.
Try it Now 1
Consider that the slope could also be written as . Find another point on the graph that has a negative value.
The Actions of the Slope and