3.2            GRAPHING EQUATIONS

Homework: Odds 1-43

 

A. Solutions of Equations.

Equations having more than one variable, such as y = 3x +2, have solutions that are ordered pairs, (x, y). These ordered pairs contain the corresponding x and y-values that make the equation a true statement.

 

Ex.  For y = 3x +2, the ordered pair (-5, -13) is a solution. This can be verified by substituting the x-value of –5 and the y-value of –13 from the ordered pair in place of the x and the y in the equation, as shown:

 

Y = 3x + 2

-13 = 3(-5) + 2       Substitute the x and y-values into the equation.

-13 = -15 + 2          Perform the calculations on the right side.

-13 = -13                Verify that a true statement results.

 

Practice Problem:

Determine whether the ordered pairs, (-6, -12) and (5, 1) are solutions of the equation,

4x – 3y = 12.

 

 

B. Graphing Linear Equations.

 

Linear equations are equations of the forms  y = mx + b  or  ax + by = c. When the solutions of these equations, the ordered pairs, are graphed, these points form a straight line.

 

Graphing By Hand:

1.       Choose a value to substitute for x.

2.     Calculate to find the corresponding y-value for the chosen x-value.

3.     Write these values as an ordered pair.

4.     Repeat the process to find at least one other ordered pair.

 

 

 

 

Ex.  Graph y = 2x – 5 by hand.

 

First make a table to find ordered pairs that are solutions of the equation. Remember that the x-values are chosen first. Calculate each x-value in the equation to find the corresponding y-value.

             X

Y_________________                                  

Y = 2x – 5

            (x, y)

            -1

Y = 2(-1) – 5 = -2 – 5 = -7

           (-1, -7)

             0

Y = 2(0) – 5 = 0 – 5 = -5

           (0, -5)

             1

Y = 2(1) – 5 = 2 – 5 = -3

           (1, -3)

 

Draw the graph of the equation by plotting the ordered pairs and then drawing a straight line through the points.

 

 

Practice Problem:

Graph the linear equation  y = - x + 3  by hand.

 

 

Graphing with grapher.

Enter the equation that you want to graph into the calculator in the Y= window.

 

Graph y = 3x using grapher.

                              

 

The TRACE feature shows coordinates of points on the line. As you move the cursor on the line using the left or right arrows, the x and y coordinates of each point are displayed at the bottom of the graph.

 

The TABLE feature shows the ordered pairs, or solutions, of the equation.

 

C.  Graph Nonlinear Equations.

 

a)

         

         

 

b)

c)

 

 

D.  Determine an Appropriate Viewing Window.

 

If your viewing window is not appropriate to the function, then parts or even all of the graph may not appear in the graphing window. To see farther up, increase the y-max, to see farther down, decrease y-min. To see farther left, decrease x-min, and to see farther right, increase x-max.

Another option that helps us to find a good viewing window is ZoomFit.

The standard viewing window, Zstandard, is shown below:

 

 

 

Practice Problem:

Find an appropriate viewing window for the graph of  y = x + 25.

 

Hint: Increase the y-max, since the y-intercept of the graph is found at

(0, 25).