Graphing Method #3:  A Point & A Slope

If you have two points you can draw a line.  So to graph a linear equation you just need an initial point, and a direction and distance to count.  That would give you the two points.

For example, you know that (3, -2) was a solution to the equation in question.  Let’s say you also knew the slope was 1/2.  You’d plot the point (3, -2), then count up one, and to the right two.  Because the slope of ½ is positive, you want to make sure that the direction of the line, when read from left to right, is going up.  (Note:  The scale on this coordinate plane is that each line is ½, so be careful when counting!  Check the scale.)

There are a lot of ways to find a point and the slope.  Sometimes you can figure both out easily, as in an equation like 3x – y = 12.  Here the x – intercept is (4, 0), and the slope is 1/3.  (Remember, –B/A is the slope when written in Standard Form.)

Perhaps the easiest way to graph a line is from Point-Slope form.  That is y = mx + b.  For example, y = -2x + 4. Remember, you just need a starting point and from which you can count your slope.  Connect the dots, boom, you’re done graphing the line.  You can always check another point on the line to see if it is a solution (good idea).

If we plugged in x = 0 to find the y – intercept, we get y = -2(0) + 4.  So, the y – intercept is (0, 4).  The slope is the coefficient of x, which here is -2.  Remember, slope is rise over run, and -2 is the same as -2/1.  So, down two, then right one.

Note:  When counting a negative slope, you can either:

• Count down and then right
• Count up and then left

So, to graph from Point-Slope formula:

1. Plot the y – intercept (0, b)
2. Count the slope to find the nearest integer coordinate
3. Connect the dots (conecta los puntos, as they say in Spanish)

Let’s summarize what we’ve learned about graphing.

• A graph is a picture of all the solutions.
• The solutions to linear equations are co-linear (form a line).
• A t-chart ALWAYS works and should be used if you get stuck.
• y = # is a horizontal line.
• x = # is a vertical line.
• Graphing by finding the intercepts is very useful and easy.
• You only need two lines to graph a linear equation.
• Point – Slope form provides you a point and slope.

Big Idea

There are infinitely many solutions to linear equations (equations with a largest exponent of 1, basically).  A graph allows us to visually examine and understand what the equation describes.

When graphing you’re drawing a picture of all solutions.  You only need two solutions to make a graph of a line!  All methods just help you find two points (at a minimum).