Graphing is the easiest method to find the slopes, character of the function/equation, vertex, intercepts, so on. Graphing concept not only helps Algebra to do problems faster but also helps in Geometry, Trigonometry, Statistics, etc.

Graphing a line means using several points that the line contains to place the line correctly coordinate plane. In this section we will be learning more about algebra graphing.

The graph of a linear equation is a line. Although drawing a line is a simple geometric construction, getting a line requires knowledge of algebraic concepts.
In general any equation of the form $Ax+By = C$, where $A$, $B$, and $C$ are constants and $x$ and $y$ are variables, is a linear equation in two variables. The knowledge that any equation of the from $Ax+By = C$ produces a straight line graph, along with the fact that two points determine a straight line, makes graphing linear equations in two variables a simple process.
Any of the following methods can be used to obtain points that satisfy a linear equation whose graph is an straight line.
  1. Intercept method: Determine from the linear equation the point at which the line crosses the x-axis by letting y = 0 and solving for x, and the point at which the line crosses the y-axis by letting x = 0 and solving for y. This method can be used only for oblique lines that do not pass through the origin. Example: The graph of the equation $2x + 3y = 6$ crosses the x-axis at (3,0), and the y-axis at (0,2).
  2. Slope-Intercept method: Put the equation in $y = mx+b$ form. Graph (0,b), the point at which the line crosses the y-axis. Use the slope to locate at least one additional point.
  3. Three-point method: Solve the given equation for y, if necessary. Determine the coordinate of three points on the line by replacing x with three convenient values and then determining the corresponding values of y.

To graph a two- variable linear inequality, first graph the corresponding equation. Then determine which side of the line contains the set of points that satisfy the original inequality.
  1. If the coordinates of the test points satisfy the inequality, the the solution contains all the points on the same side of the line. Otherwise, the solution contains all the points on the opposite side of the line.
  2. If the inequality relation is $<$ or $>$, draw a dashed boundary line.
  3. If the inequality relation is $\leq$ or $\geq$, draw a solid boundary line which indicates that points on the line are also included in the solution.
The following are the examples of algebra graphing.

Solved Examples

Question 1: Solve the given equation for y; y = 2x+4. And find the corresponding values of y for three convenient values of x, say x = -2,0,2.
Solution:
 
Given  y = 2x+4
Let x = -2 ; 
y = 2(-2)+4 = 0
The point (-2,0) is a solution.
 
Let x = 0 ; 
y = 2(0)+4 = 4
The point (0,4) is a solution.

Let x = 2 ; 
y = 2(2)+4 = 8
The point (2,8) is a solution.
 
The graph can be plotted as below.
Graphing Linear Equations

 

Question 2: Graph $2x-3y = 6$
Solution:
 
We recognize that equation 2x-3y = 6 is a linear equation in two variables, therefore its graph will be a straight line.
Let x = 0: then 2(0) – 3y = 6
-3y = 6
y = -2
 thus (0,-2) is a solution.

Let y = 0: then 2x -3(0) = 6
2x = 6
x = 3

Thus (3,0) is a solution

Let x = -3: then 2(-3)-3y =  6
-6-3y = 6
-3y = 12
y = -4
thus (-3, -4) is a solution 

by plotting the points associated with these three solutions  and connect them with a straight line to produce the graph of $2x-3y=6$

Linear Equation Graphing
 

Question 3: Graph the solution $2x-y \leq 3$
Solution:
 
Solve the inequality for y.
  $2x-y \leq 3$
$-y \leq -2x+3$
$y \geq 2x-3$
graph y = 2x-3 as a solid line. Because the inequality is $\geq$, shade above the line. 

Algebra Graphing