Thursday, April 23, 2015

Graph Lines Using Intercepts

The x-intercept is the point where the graph intersects the x-axis and the y-intercept is the point where the graph intersects the y-axis. These points have the form (x, 0) and (0, y) respectively.
x and y-intercepts
To find the x- and y-intercepts algebraically, we use the fact that all x-intercepts have a y-value of zero and all y-intercepts have an x-value of zero.  For example,
Graph:  3x − 5y = 15
Tip 1: To find the y-intercept, set x = 0 and determine the corresponding y-value.  Similarly, to find the x-intercept we set y = 0 and determine the corresponding x-value.
 
Keep in mind that the intercepts are ordered pairs and not numbers.  In other words, the x-intercept is not x = 5 but rather (5, 0).

Two points determine a line. If we find the x- and y-intercepts, then we can use them to graph the line. As you can see, they are fairly easy to find. Plot the points and draw a line through them with a straightedge.
Done. Let’s do another one.
Graph: yx + 9
We begin by finding the x-intercept.
The x-intercept is (3, 0).
The y-intercept is (0, 9). Now graph the two points.
Graph of the Line
Tip 2: Use Desmos.com to check your answer – it’s totally free.  Just type in the equation.

This is a nice and easy method for determining the two points you need for graphing a line.  In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.

Graph −4x + 3y = 12 using the intercepts.

  

Graph −4x + 2y = −6 using the intercepts.

  

Graph  y = −5x +15 using the intercepts.

   

Graph  y = −3/4 x + 9 using the intercepts.


This brings us to one of the most popular questions in linear graphing.  Do all lines have x- and y-intercepts?  The answer is NO.  Horizontal lines, of the form y = b, do not necessarily have x-intercepts.  Vertical lines, of the form x = a, do not necessarily have y-intercepts.

Graph y = 3.

  

Graph x = −2.

Many students this method, but I will tell you, there is a better way. Even less work... [ Graph Lines using Slope and Intercepts ] Read on!
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