Methods for Graphing Linear Functions (part I)
Method 1: Graphing by finding ordered pairs.
One way to graph a linear function is to find two points satisfying your equation.
A point (x, y) satisfies your equation if the equation is true when you plug in your
point. For example the point (2, 1) satisfies the equation
because
( )
To find points that satisfy our equation we can use several strategies. Begin by
looking at the way the equation looks. If it is in slope-intercept form
ie.“y = something”, plug values in for x to find y. You can choose any value you
want for x, but go with something that makes it easy for you to simplify the
equation like x = 0 and x = 1.
Example 1a: Find two points satisfying the equation
.
Using x = 0, the point would be (0, -3)
Using x = 1, the point would be (1, -1)
Now that you have found 2 points satisfying your line you can graph your function
by plotting those points and drawing a line through the points you have found. If
you found the points correctly any other point satisfying the equation will fall on
the line you draw. For example (2, 1) would fall on the line you draw through
(0, -3) and (1, -1).
Example 1b: Graph the linear function
.
Graph of example 1:
Unfortunately x = 0 and x = 1 are not always the best choices to find y.
Example 2a: Find 2 points satisfying the linear function
.
x = 0 is still a good choice because the fraction goes away when you plug in 0 for
x. What is the point you get when plugging in 0? (0, -3)
x = 1 isn’t a great choice because there is still a fraction when you plug in 1. Is
there a value you can plug in to get rid of fraction?
To answer that question we have to figure out something that the denominator of
the fraction will go into. So what is the smallest thing 5 will go into? 5
So let’s try x = 5. What is the point you get when plugging in x =5? (5, 0)
When x = 5,
( )
(the 5’s cancel)
If we wanted a 3rd point satisfying equation, what are some values we might
choose for x? 10, 15 etc. any multiple of 5 would be ok.
Example 2b: Graph the function
.
Graph of Example 2:
Notice: The points that we graphed fall on the x and y-axis. The point (0, -3) falls
on the y-axis so we call this point the y – intercept. The point (5, 0) falls on the
x-axis so we call this point the x -intercept. If the equation is not in slope
intercept form, a good way of graphing is to find the intercepts of the function.
Consider the following example.
Example 3a: Find the x and y-intercepts of the function
To find the x-intercept we let y = 0?
To find the y-intercept we let x = 0?
.
( )
Plugging in y = 0,
The x-intercept is (3 , 0). Visually you can cover-up the y-term and solve for x.
Plugging in x = 0,
( )
The y-intercept is (0, -2 ). Visually we can cover-up the x-term and solve for y.
Example 3b: Graph
using the x and y-intercepts.
Ex. 4a Find three points satisfying the equation
.
This equation will be solved when y = 3. Notice that this equation does not
depend on x at all. As long as the y – value of your ordered pair is 3, our point will
satisfy the equation.
So find three x-values to complete the following three ordered pairs.
( 0, 3) , ( 1 , 3), ( 2 , 3)
Ex. 4b Graph the function
using the 3 points you found in a.
Answers will vary but the graph will look the same regardless of the 3 points that
you used.
*If you can put your equation in the form
get a horizontal line.
where c is a constant you will
Ex 5a: Find three points satisfying
.
This equation will be solved when x = -3. So find the missing values for each of
the following ordered pairs. ( -3, 0) , ( -3 , 1) , ( -3 , 100)
Ex 5b: Graph the function
using the first 2 points you found above.
*If you can put your equation in the form
a vertical line.
where c is a constant you will get