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 Point-Plotting Method. 

Point-Plotting Method

Examples

Determine the following points are solutions to the given equation.

1.   y=\sqrt{x+4}            a)  \left ( 0,2 \right )                            b)  \left ( 5,3 \right )

                                         2 =\sqrt{0 + 4}                           3=\sqrt{5+4}

                                        2=\sqrt{4}                                    3=\sqrt{9}

                                        2=2                                       3=3

                                        yes                                            yes

 

2.  y=4-\left | x-2 \right |               a)  \left ( 1,5 \right )                      b)  \left ( 6,0 \right )

                                                5=4-\left | 1-2 \right |                0=4-\left | 6-2 \right |

                                               5=4-\left | -1 \right |                     0=4-\left | 4 \right |

                                               5=4-1                           0=4-4

                                              5=3                                   0=0

                                              no                                         yes

 

There are limitations to the point-plotting 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:

  1. what type of equation you have

  2. what the intercepts are if any

  3. 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.  y=16-4x^{2}

Type of Graph:   This is a parabola that opens down. 

The intercepts are:

x-intercept:      \left ( x,0 \right )    \rightarrow \; 0=16-4x^{2}

                                                              4x^{2}=16

                                                               x^{2}=4

                                                               x=\pm 2

                                   The x-intercepts are \left ( -2,0 \right ) and \left ( 2,0 \right ).

y-intercept:    \left ( 0,y \right )     \rightarrow \; y=16-0

                                                y=16

                                   The y-intercept is \left ( 0,16 \right ).

Now check for symmetry:

x-axis symmetry:  Replace y with -y  \rightarrow \; -y=16-4x^{2}    no

y-axis symmetry:  Replace x with -x  \rightarrow \; y=16-4\left ( -x \right )^{2}

                                                                            y=16-4x^{2}     yes

Since we know this is a parabola that opens down and it is symmetrical about the y-axis it is very easy to sketch the graph plotting the three intercepts that we found.   

2.  y=x^{3}-4x

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.

x-intercepts:  \left ( x,0 \right )    \rightarrow \; 0=x(x^{2}-4)

                                              0=x\left ( x+2 \right )\left ( x-2 \right )

                                        The x-intercepts are  \left ( 0,0 \right ), \left ( 2,0 \right ) and \left (- 2,0 \right ).

y-intercepts:  \left (0,y)\right   \rightarrow \; y=0

                                        The y-intercept is \left ( 0,0 \right ).

Now check for symmetry:

x-axis symmetry:  Replace y with -y  \rightarrow -y=x^{3}-4x   no

y-axis symmetry:  Replace x with -x  \rightarrow y=\left ( -x \right )^{3}-4\left ( -x \right )

                                                                           y=-x^{3}+4x  no

origin symmetry:  Replace (x,y) with (-x,-y)  \rightarrow \; -y=-x^{3}+4x

                                                                                               y=x^{3}-4x  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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