Intercepts

There are some key features of graphs that we can use to help us create graphs more quickly and easily.  The intercepts are absolutely key points on a graph, and meaningful algebraically, as well.  A function will never have more than one y – intercept, but can have many x – intercepts.  However, linear equations have only one x – intercept. 

Here are some key pieces of information about intercepts.

  1. An intercept is where the graph crosses an axis.
    1. There are two axes, the x, which is horizontal and the y, which is vertical.
  2. For all x – intercepts, the y – coordinate is 0.
    1. To find an x – intercept, replace y with zero and solve for x.
  3. For all y – intercepts, the x – coordinate is 0.
    1. To find a y – intercept, replace x with zero and solve for y.

Because it is easy to do math with zeros, it is a good idea to find the intercepts when using a t-chart!

Universal:  Each type of function and equation has its own patterns and short cuts.  In all of them, the values of x and or y being zero are of particular importance.  The intercepts play key roles and are almost always one of the first pieces of information to be found.  With linear equations, we frequently find the y – intercept because it is easy to do so, frequently does not require any calculation at all. Our focus here is linear equations, but the process is the same for all equations.  Let’s do a few examples.

Find the intercepts of 4x – 3y = 24.

x – intercept:  This will be a coordinate (#, 0).  So, y will be zero.  Replace y with zero in the equation, then solve for x.

Note:  This is not the x – intercept, but the x – coordinate of the x – intercept.  The intercept is (6, 0).

y – intercept: This will be a coordinate (0, #).  So, x will be zero. Just replace x with zero and solve for y.

Note:  This is not the x – intercept, but the x – coordinate of the x – intercept.  The intercept is (6, 0).

y – intercept: This will be a coordinate (0, #).  So, x will be zero. Just replace x with zero and solve for y.

Note:  This is not the y – intercept, but the y – coordinate of the y – intercept.  The intercept is (0, -8).

If you know the two intercepts, you can sketch a graph of the line!  Here we know (6, 0) and (0, -8) are the intercepts.  If we plot those two points, we can connect the dots and have a graph.  We might wish to check a random point on the line, plug in the x and y to see if it is a solution.  But, if after checking we find a third point is a solution, we can be sure our graph is correct.

This is Graphing Method #2 of graphing a line, Finding the Intercepts.

As has proven to be the case with linear equations, there are a few exceptions.  There are three times when this method does not provide quite enough information.  You can probably guess two of them, the horizontal and vertical line.  Here’s how to handle those equations.

Graphing a Vertical Line

If you’re asked to graph x = #, recognize it will be a vertical line.  Every coordinate on the line will have an x component that is the #.  So, for x = 14, all coordinates will be (14, some number for y).  Just place two points with an x – coordinate of 14, and connect the dots.  Extend your line to each end of the coordinate plane.

There will NOT be a y – intercept for a vertical line.  However, if the line is x = 0, it will be the y – axis.

Graphing a Horizontal Line

If you’re asked to graph y = #, recognize that it will be horizontal.  Every coordinate on that line will have a y component that is the #.  So, if you had y = 81, every coordinate would be (some number for x, 81).  Just place two points on the with a y – coordinate of 81, and connect the dots.  Extend the line to each end of the coordinate plane.

There will NOT be an x – intercept for a horizontal line. However, if the line is y = 0, it will be the x – axis. 

The Origin

The third time finding the intercepts of a linear equation fails to provide enough information to make a graph is if the intercepts pass through the origin.  In this case, you’ll need to either use a t-chart, count your slope (you’ll see that in the next section), or just find another point (explained soon).

Here’s an example: 

y = 2x

The x – intercept is (0, 0).  Here’s the work:  (0) = 2xx = 0.

The y – intercept is (0, 0).   Here’s the work:  y = 2(0) → y = 0. 

Even though we found both intercepts, they’re at the same exact spot.  We need two points to draw a line (without guessing its direction).  In this case, probably the easiest thing to do is just figure out another solution.  The equation says that the output (y) is twice as big as the input (x).  To make them equal, the x is doubled.  So, if x was 3, y would be 6.  (3, 6) will be a point on the line.  Plot (0, 0), and then (3, 6), and you’re done!

Summary:  A graph is a picture of all solutions to an equation.  If you know the shape of the equation (linear equations form a line), then you only need a few clues as to where the graph will be.  A t-chart is a sure-fire, always works, way to find out information for any equation!  The next method, that is almost as reliable, and will translate to almost any type of graph is finding the intercepts.  For an x – intercept, y = 0, and for a y – intercept, x = 0. 

There are two strange cases with linear equations, the vertical and horizontal lines.  The line y = 3, for example, is horizontal because every coordinate on that line will be some number for x and y = 3, like (2, 3), (0, 3) and (-11, 3).  It will have a y – intercept, but not an x – intercept.

A vertical line will be x equals a number, like x = 3.  This will have an x – intercept, but not a y – intercept.  Every coordinate on this line will have x = 3, and y can be any number. 

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