The Graph of an Equation
Monday, April 9th, 2012 at
12:16 pm
The Graph of an Equation
The graph of an equation in two variables can be displayed on a coordinate plane. All ordered pairs (a,b) that are solutions of the equation are points on the graph. The graph of the equation is a set of all points that are solutions of the equation.
To sketch a graph you can use a number of methods depending on the type of graph you have. One method is called the PointPlotting Method.
PointPlotting Method

make a table of values

plot the points

connect the points
Examples
Determine the following points are solutions to the given equation.
1. a) b)
yes yes
2. a) b)
no yes
There are limitations to the pointplotting method as it is often difficult to determine points that will be solutions to the equation. How do we know if the graph is a straight line or a curved path? Is this equation a parabola or a circle?
To sketch a graph it is helpful to know the following sketching aids:

what type of equation you have

what the intercepts are if any

what kind of symmetry (if any) that the graph has
Once you know the above you can draw a sketch of any equation.
Examples:
1.
Type of Graph: This is a parabola that opens down.
The intercepts are:
xintercept:
The xintercepts are and .
yintercept:
The yintercept is .
Now check for symmetry:
xaxis symmetry: Replace y with y no
yaxis symmetry: Replace x with x
yes
Since we know this is a parabola that opens down and it is symmetrical about the yaxis it is very easy to sketch the graph plotting the three intercepts that we found.
2.
Type of graph: This is a cubic equation with a positive leading coefficient so the graph starts from negative infinity and moves from left to right ending up at positive infinity.
xintercepts:
The xintercepts are , and .
yintercepts:
The yintercept is .
Now check for symmetry:
xaxis symmetry: Replace y with y no
yaxis symmetry: Replace x with x
no
origin symmetry: Replace (x,y) with (x,y)
yes
Since we know this is a cubic equation that falls to the left and rises to the right, and we know it is symmetric about the origin, we can use the three intercepts and easily sketch the graph of the equation.
Filed under: Functions and Their Graphs
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