Graphing Linear Equations and Inequalities (Homer Simpson)

Homer1The following examples come from RegentsPrep.org

Practice Problems:

Slope and Graphing

Graphing Inequalities

 

When working with straight lines, there are several ways to arrive at an equation which represents the line.   

Remember:

Slope is found by using the formula:
 

Slope is also expressed as rise/run.

Equation Forms of Straight Lines

Slope Intercept Form

Point Slope Form

Use this form when you know the slope and they-intercept (where the line crosses the y-axis).

= mx + b

m = slope
y-intercept
(where line crosses the y-axis.)

Use this form when you know a point on the line and the slope (or can determine the slope).

m = slope

= any point on the line

Horizontal Lines

Vertical Lines

y = 3 (or any number)
Lines that are horizontal have a slope of zero. Horizontal lines have “run”, but no “rise”.   The rise/run formula for slope always yields zero since the rise = 0.
Since the slope is zero, we have
= mx b
= 0•x + 3
y = 3
This equation also describes what is happening to the y-coordinates on the line.  In this case the y-coordinates are always 3.

x = -2 (or any number)
Lines that are vertical have no slope (it does not exist).  Vertical lines have “rise”, but no “run”.  The rise/run formula for slope always has a zero denominator and is undefined.

The equations for these lines describe what is happening to the x-coordinates.  In this example, the x-coordinates are always equal to -2.


Examples:

Examples using Slope-Intercept Form:

Examples using Point-Slope Form:

1.  Find the slope and y-intercept for the equation 2y = -6x + 8.

First solve for “y =”:      y = -3x + 4
Remember the form:     mx + b
Answer:  the slope (m) is -3
the y-intercept (b) is 4
3.  

2. Given that the slope of a line is -3 and the line passes through the point (-2,4), write the equation of the line. The slope:  m = -3
The point (x,y1) = (-2,4)
Remember the form:  y - y1 = m ( x - x1)
Substitute:                 - 4 = -3 (x - (-2))
ANS.                        y - 4 = -3 ( x + 2)

If asked to express the answer in “y =” form:

y - 4 = -3- 6
y = -3x - 2 
2.  Find the equation of the line whose slope is 4 and the coordinates of the y-intercept are (0,2).

In this problem m = 4 and b = 2.
Remember the form:  y = mx and that bis where the line crosses the y-axis.
Substitute:   y = 4x + 2    

3.  Find the slope of the line that passes through the points (-3,5) and (-5,-8).

First, find the slope:    


Use either point:  (-3,5)
Remember the form:  y1 = m ( x - x1)
Substitute:  y - 5 = 6.5 ( x - (-3))
- 5 = 6.5 (x + 3)  Ans.

There are several ways to graph a straight line given its equation.   

Let’s quickly refresh our memories on equations of straight lines:

Slope Intercept Form Point Slope Form Horizontal Lines Vertical Lines


when stated in “y=” form, itquickly gives the slope, m, and where the
line crosses the y-axis, b,called the y-intercept.


when graphing, put this equation into “y = ” form to easily read graphing information.

y = 3 (or any number)
horizontal lines have a slope of zero – they have “run”, but no “rise” — all of the y values are 3.

= -2 (or any number)
vertical lines have no slope (it does not exist) – they have “rise”, but no “run” –all of the x values are -2.

Graphing Tidbits:

If a point lies on a line, its coordinates make the equation true.

(2,1) in on the
line = 2- 3
because 1 = 2(2) – 3

Before graphing a line, be sure that your equation starts with “y=”.

To graph 6+ 2y = 8
rewrite the equation:
2= -6+8
  y = -3+ 4
Now graph the line using either slope intercept method or chart method.

The x-coordinate may be called the abscissa.

The y-coordinate may be called the ordinate.

Methods of Graphing a Line

Using  y mx b
with rise/run
Using a Chart -
Plotting Points

Graph  2y = 6x + 4

1.  Put your equation in “y=” form.
         = 3x + 2

2.  The number in front of x is the slope.
(If necessary, place this number over 1 to
form a fraction for your rise/run.)
                    slope = 3/1       

3.  The “b” value is where the line crosses the
y-axis.  Be sure to check the sign of this
number.    b = 2

4.  Plot the b value on the y-axis.
see graph below

5.
  Standing at this point, use your rise and run
values to plot your second point.
(If rise is positive, move up.  If rise is negative,
move down.)
(If run is positive, move right.  If run is
negative, move left.)

6.
  Connect the two points to form the line.

Graph  2y = 6x + 4

X Y
-3
-2
-1
0
1
2
3

Create a chart to hold x and y values from your line.  For lines, the x-values usually range from -3 to +3, but may be any values you wish.

While charts often contain more than 2 entries, only two entries are needed to determine a straight line.  A third point should be used to “check” that an error was not made while computing the first two points.

X Y
-3 -7
-2 -4
-1 -1
0 2
1 5
2 8
3 11

Substitute the x-values into the equation to determine the y-values.  Putting the equation in “y=” form first will make the substitution easier.
y = 3+ 2

Now start substituting.  For example, substitute x = -3:
= 3 (-3) +2 =  -9 + 2  = -7

  Plot the (x,y) coordinates to graph the line.

If you can graph a straight line, you can graph an inequality!
  Graphing an inequality starts by graphing the corresponding straight line.  After graphing the line, there are only two additional steps to remember. 

1. Choose a point not on the line and see if it makes the inequality true.  If the inequality is true, you will shade THAT side of the line — thus shading OVER the point.  If it is false, you will shade the OTHER side of the line — not shading OVER the point.
2. If the inequality is LESS THAN OR EQUAL TO or GREATER THAN OR EQUAL TO, the line is drawn as a solid line.  If the inequality is simply LESS THAN or GREATER THAN, the line is drawn as a dashed line.

Graphing an Inequality

1.  Solve the equation for y (if necessary).
2.  Graph the equation as if it contained an = sign.
3.  Draw the line solid if the inequality is   or 
4.  Draw the line dashed if the inequality is 
< or >
5.  Pick a point not on the line to use as a test point.
The point (0,0) is a good test point if it is not on
the line.
6.  If the point makes the inequality true, shade that
side of the line.  If the point does not make the
inequality true, shade the opposite side of the line.

Graph the following  inequality 

y  3x - 1

Graph the inequality
y 
 3x - 1

1.  Graph the line = 3x - 1.
2.  Pick a test point.  (0,0) was used.

3.  The test point is false in the inequality
                   0  3(0) – 1
 0  -1   false
4.  Since the test was false, d
o not shade OVER the point (0,0) — shade the opposite side of the line.
5.  The line, itself, is SOLID because this problem is “less than or EQUAL TO.”

Additional Resources

Graphing Linear Equations

Graphing Linear Inequalities

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