Graphing Formulas

Points that satisfy a formula

Given a formula in two variables x and y in a Cartesian coordinate system, a point (a , b) is said to satisfy the formula if substituting x = a and y = b into the formula produces a true expression.

Steps to determine if (a,b) satisfies a formula

Step 1: Substitute x = a and y = b into the formula.
Step 2: Simplify
Step 3: Determine whether the resulting expression is true.

  • If yes, the point (a , b) satisfies the formula.

  • If no, the point (a , b) does not satisfy the formula.


Examples

Example 1

Does (2 , -3) satisfy the formula y = 1 - x2 ?

Step 1: Substitute x = 2 and y = -3 into the formula.

-3 = 1 - (2)2,

Step 2: Simplify

-3 = 1 - 4,

Step 3: Determine whether the resulting expression is true.

Yes.


Conclusion: The point (1 , 0) does satisfy the formula


Example 2

Is (3 , -2) a solution for the formula y - 3x - 5 = 0 ?

Step 1: Substitute x = 3 and y = -2 into the formula.

-2 - 3(3) - 5 = 0,

Step 2: Simplify

-16 = 0

Step 3: Determine whether the resulting expression is true.

No.


Conclusion: The point (3 , -2) does not satisfy the formula




Definition

A graph is the set of all points (x , y) in the xy plane that satisfy a given formula.

Steps to find the graph of a formula

Step 1: Construct a table of x values.

x -2 -1 0 1 2
y

Step 2: For each x-value find the y-value that satisfies the formula.
Step 3: Complete the table.
Step 4: Plot all of these points in the xy plane

Click here if you have problems with Step 4.
Step 5: Can you guess the curve that all points will form?

  • If yes, draw the graph (place the appropriate curve through the given points).

  • If no, return to Step 1 and add more values of x to the table.

Examples

Example 1

Graph: y = -2x -1

Step 1: Construct a table of x values.

x -2 -1 0 1 2
y

Step 2: For each x-value find the y-value that satisfies the formula.

If x =-2 , then y = -2(-2) -1 = 3

If x =- 1 , then y = -2(-1) -1 = 1

If x =0 , then y = -2(0) -1 = -1

If x =1 , then y = -2(1) -1 = -3

If x =2 , then y = -2(2) -1 = -5
Step 3: Complete the table.
x -2 -1 0 1 2
y 3 1 -1 -3 -5

Step 4: Plot all of these points in the xy-plane.

Step 5: Can you guess the curve that all points will form?

  • yes, this is a line.

Example 2

Graph: y = x2 - 2

Step 1: Construct a table of x values .

x -2 -1 0 1 2
y

Step 2: For each x-value find the y-value that satisfies the formula.

If x =-2 , then y = (-2)2 - 2 = 2

If x =- 1 , then y = (-1)2 -2 = -1

If x =0 , then y = (0)2 - 2 = -2

If x =1 , then y = (1)2 - 2 = -1

If x =2 , then y = (2)2 - 2 = 2
Step 3: Complete the table.
x -2 -1 0 1 2
y 7 4 3 4 7

Step 4: Plot all of these points in the xy-plane.

Step 5: Can you guess the curve that all points will form?

  • yes, this is a parabola.


Graphing Other Expressions


To see the graphs of other expressions, enter expressions which only contain the variable x in the text box. When you click on graph it, you will see the graph(s) of y = expression where expression has been entered in the text box.

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