Topic 4.1 (Graphing Equations) 

There are three ways to graph a linear equation (a straight line).

1.     Using a table of values.

2.    Using the x- & y-intercept method(also called cover up).

3.    Using the slope, y-intercept equation(y = mx + b).

                m = slope,            b = y-intercept

            Slope   =   rise/run   =   (change in y)/(change in x)   =   (y₂ - y₁)/(x₂ - y₁)

            When an equation has a positive slope it will slant up to the right.

             When an equation has a negative slope it will slant down to the right.

             The slope of a horizontal line is 0.

             The slope of a vertical line is undefined.

             Two Parallel lines have the same slope.

             Two Perpendicular lines have slopes that are negative reciprocals.

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Sample Problems

Graph the equation:       4x + y = 12       solve for y:        y = -4x + 12

    (use a table of values)      x       y          substitute values for x & solve for y.

                                                   -3      24         y = -4(-3) + 12,  y = 12 + 12,  y  =  24 

                                                   -2      20        

                                                   -1      16          

                                                    1         8         Plot at least two points, three is

                                                    2         4         better.

                                                    3         0

    The graph is this equation will go through the points:  (1,  8), (2, 4), & (3, 0).

Draw the line between the three points and The line will slant down to the right.

(This means the graph has a negative slope.)

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Graph the equation:               5x - 4y = 20

         (use x- & y-intercept)              x            y

                                                               4            0        Make x or y = to 0 and solve for

                                                               0           -5        the variable that’s left.

           Plot the two points.

     Draw the line between the x & y intercept and the graph will slant up to the

right.  (This means the graph has a positive slope.)

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Graph the equation:                     x + 4y = 4

            (use the slope, y-intercept equation)        (y = mx + b)

                                    4y  =  -x  +  4                               Solve the equation for y.

                                      y = (-1/4)x + 1

                   (-1/4) - (m) slope        1 - (b) y-intercept

     Plot the y-intercept(1) and use the slope to determine another point. 

                Remember the slope is (y distance)/(x distance)

     To graph the equation, start at the y-intercept (0, 1).  Then according to the

slope (-1/4), move the line 4 units to the right and 1 unit down.  The point you are

now is (4, 0).  Draw the line between the two points.  The graph slants down to

the  right.

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Graph the equation:                        x    =    4

                         When the linear equation has only an x, the graph is a vertical line

                              passing through the constant on the x axis.

   Put a dot on 4 on the x-axis and draw a vertical line through it into quadrants

I  &  IV.

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Graph the equation:                     y    =    1

                         When the linear equation has only a y, the graph is a horizontal line

                               passing through the constant on the y axis.

     Put a dot on 1 on the y-axis and draw a line horizontal through it into quadrants

I  &  II.

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Find the x- and y-intercepts of the line:         x   -   (3/4)y   =   2

     Substitute 0 for y and solve for x:    x   -   0   =   2,   x = 2,    (2,0)  x-intercept

     Substitute 0 for x and solve for y:     0   -   (3/4)y   =   2  Multiply both sides by 4.

                                                                               3y = 8,    y = 8/3,   (0, 8/3)  y-intercept

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Find the slope of the line through        (2, 4)     &     (5, 7)

Use the slope formula:        m =(y₂ - y₁)/ (x₂ - x₁)  =   (7 - 4)/(5 - 2)  =  3/3 =   1 - Slope

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Find the slope of a line perpendicular to the line through the points

   (2, -4) & (6. 1).         Find the slope of the line first that has the points

                       given. m = [1 - (-4)]/(6 - 2) = (1 + 4)/(6 - 2)  =      5/4 -  Slope

         The answer is   -4/5.

To find the slope of a line perpendicular you must take the negative reciprocal

of the original slope.  (5/4)  becomes  (-4/5)

If the you wanted the slope of the line parallel to the line through the

given points, just use the same slope.