Category Archives: Graphing

2.2 Circles, 2.3 Circles, Finding Intercepts of a Circle, Intercpets

Finding the Intercepts of a Circle (4 Intercepts)

An x-intercept is where the graph touches or crosses the x-axis.

A y-intercept is where the graph touches of crosses the y-axis.

To find an x-intercept: Let y=0 and solve for x.

To find an y-intercept: Let x=0 and solve for y.

 

Example:  Find the intercepts of the circle for the given equation.

(x+3)^2+(y+6)^2=81

Solution:

To find an x-intercept, let y=0 and solve for x.

(x+3)^2+(y+6)^2=81

(x+3)^2+(0+6)^2=81

(x+3)^2+(6)^2=81

(x+3)^2+36=81

(x+3)^2+36-36=81-36

(x+3)^2=45

sqrt{(x+3)^2}=sqrt{45}

x+3=pm sqrt{9*5}

x+3=pm 3sqrt{5}

x+3-3=-3 pm 3sqrt{5}

x=-3 pm 3sqrt{5}

Approximately x=-9.708 and x=3.708

This equation has two x-intercepts. (-3 + 3sqrt{5},0) and (-3 - 3sqrt{5},0)

To find a y-intercept, let x=0 and solve for y.

(x+3)^2+(y+6)^2=81

(0+3)^2+(y+6)^2=81

(3)^2+(y+6)^2=81

9+(y+6)^2=81

(y+6)^2+9-9=81-9

(y+6)^2=72

sqrt{(y+6)^2}=sqrt{72}

y+6=pm sqrt{36*2}

y+6=pm 6sqrt{2}

y+6-6=-6 pm 6sqrt{2}

y=-6 pm 6sqrt{2}

Approximately y=-14.49 and x=2.485

This equation has two y-intercepts. (0,-6 + 6sqrt{2}) and (0,-6 - 6sqrt{2})

2.1 Graphing, Graphing By Plotting Points

Graphing an Absolute Value Equation Shifted Horizontally by Plotting Points

Plotting Points Method

  1. Find ordered pair solutions
    • Choose values for one variable
    • Find the value for the other variable
  2. Plot the ordered pair solutions
  3. Draw the line or curve that connects the ordered pairs

Example: Graph  y=delim{|}{x-2}{|} by the plotting points method.

1. Find ordered pair solutions.  We can organize this information in a table.

  • Choose values for one variable

When the equation is written in “y=” format it is easier to choose values for x and then find the y’s.  Although you are allowed to choose any x values, for this table I have used a standard set of x’s so that we have a variety.  Some negatives, some positives and zeros.

xy
-2
-1
0
1
2
  • Find the values for the other variable

To find the y’s substitute the value of x into the equation and simplify to find y.

xyyy
-2|-2-2|=|-4|=4
-1|-1-2|=|-3|=3
0|0-2|=|-2|=2
1|1-2|=|-1|=1
2|2-2|=|0|=0

2.  Plot the ordered pair solutions

Using the table above we have 5 ordered pairs (-2, 4), (-1, 3), (0, 2), (1, 1), (2, 0)

Plot the ordered pairs using the rectangular coordinate system.

3.  Draw the line or curve that connects the ordered pairs.

When you look at the plotted ordered pairs you should see a pattern the that points make.  As I discussed in a previous post, most absolute value graphs form a “v” shape. It seems I am not graphing the portion of the graph that shows the “v” shape.  With the plotting points method all I can do is find more ordered pairs to try to get the correct graph.  However, in a future post, we will discuss other graphing methods give us more information about where the interesting part of the graph is.

In the mean time, to finish this graph I need just one more ordered pair.  (3, 1) which I get from plugging three into the equation for x.

This makes the graph have the “v” shape we expected.

The line represents all of the ordered pairs that are solutions to the equation y=delim{|}{x-2}{|}.

 

2.1 Graphing, Graphing By Plotting Points

Graphing an Absolute Value Equation by Plotting Points

Plotting Points Method

  1. Find ordered pair solutions
    • Choose values for one variable
    • Find the value for the other variable
  2. Plot the ordered pair solutions
  3. Draw the line or curve that connects the ordered pairs

Example: Graph  y=delim{|}{x}{|}+2 by the plotting points method.

1. Find ordered pair solutions.  We can organize this information in a table.

  • Choose values for one variable

When the equation is written in “y=” format it is easier to choose values for x and then find the y’s.  Although you are allowed to choose any x values, for this table I have used a standard set of x’s so that we have a variety.  Some negatives, some positives and zeros.

xy
-2
-1
0
1
2
  • Find the values for the other variable

To find the y’s substitute the value of x into the equation and simplify to find y.

xyyy
-2|-2|+2=2+2=4
-1|-1|+2=1+2=3
0|0|+2=0+2=2
1|1|+2=1+2=3
2|2|+2=2+2=4

2.  Plot the ordered pair solutions

Using the table above we have 5 ordered pairs (-2, 4), (-1, 3), (0, 2), (1, 3), (2, 4)

Plot the ordered pairs using the rectangular coordinate system.

3.  Draw the line or curve that connects the ordered pairs.

When you look at the plotted ordered pairs you should see a pattern the that points make.  In this case, the points form a “v” shaped graph. Most of the equations with absolute value have graphs of this shape.

The line represents all of the ordered pairs that are solutions to the equation y=delim{|}{x}{|}+2.

2.1 Graphing, Graphing By Plotting Points

Graphing a Quadratic Equation by Plotting Points

Plotting Points Method

  1. Find ordered pair solutions
    • Choose values for one variable
    • Find the value for the other variable
  2. Plot the ordered pair solutions
  3. Draw the line or curve that connects the ordered pairs

Example: Graph  y= x^2-5 by the plotting points method.

1. Find ordered pair solutions.  We can organize this information in a table.

  • Choose values for one variable

When the equation is written in “y=” format it is easier to choose values for x and then find the y’s.  Although you are allowed to choose any x values, for this table I have used a standard set of x’s so that we have a variety.  Some negatives, some positives and zeros.

xy
-2
-1
0
1
2
  • Find the values for the other variable

To find the y’s substitute the value of x into the equation and simplify to find y.

xy
-2(-2)^2-5=4-5=-1
-1(-1)^2-5=1-5=-4
0(0)^2-5=0-5=-5
1(1)^2-5=1-5=-4
2(2)^2-5=4-5=-1

2.  Plot the ordered pair solutions

Using the table above we have 5 ordered pairs (-2, -1), (-1, -4), (0, -5), (1, -4), (2, -1)

Plot the ordered pairs using the rectangular coordinate system.

3.  Draw the line or curve that connects the ordered pairs.

When you look at the plotted ordered pairs you should see a pattern the that points make.  In this case, the points form a “u” shaped curve called a parabola. Most of the equations with x squared have graphs of this shape.

The line represents all of the ordered pairs that are solutions to the equation y= x^2-5.

Here is a youtube video with a similar example.

2.1 Graphing, 2.3 Lines, Graphing By Plotting Points

Graphing a Linear Equation by Plotting Points (m is a fraction)

Plotting Points Method

  1. Find ordered pair solutions
    • Choose values for one variable
    • Find the value for the other variable
  2. Plot the ordered pair solutions
  3. Draw the line or curve that connects the ordered pairs

Example: Graph  y= -{1}/{3}x+2 by the plotting points method.

1. Find ordered pair solutions.  We can organize this information in a table.

  • Choose values for one variable

When the equation is written in “y=” format it is easier to choose values for x and then find the y’s.  For this table I will choose multiples of the denominator so that my ordered pairs are all integers.

xy
3(-2)=-6
3(-1)=-3
3(0)=0
3(1)=3
3(2)=6
  • Find the values for the other variable

To find the y’s substitute the value of x into the equation and simplify to find y.

xyyy
-6-1/3(-6)+2=2+2=4
-3-1/3(-3)+2=1+2=3
0-1/3(0)+2=0+2=2
3-1/3(3)+2=-1+2=1
6-1/3(6)+2=-2+2=0

2.  Plot the ordered pair solutions

Using the table above we have 5 ordered pairs (-6, 4), (-3, 3), (0, 2), (3, 1), (6, 0)

Plot the ordered pairs using the rectangular coordinate system.

3.  Draw the line or curve that connects the ordered pairs.

When you look at the plotted ordered pairs you should see a pattern the that points make.  In this case, the points form a straight line.

The line represents all of the ordered pairs that are solutions to the equation y= - {1} / {3}x+2.

Here is a youtube video with a similar example.

2.1 Graphing, 2.3 Lines, Graphing By Plotting Points

Graphing a Linear Equation by Plotting Points (m is an integer)

Plotting Points Method

  1. Find ordered pair solutions
    • Choose values for one variable
    • Find the value for the other variable
  2. Plot the ordered pair solutions
  3. Draw the line or curve that connects the ordered pairs

 

Example: Graph  y= 2x-5 by the plotting points method.

1. Find ordered pair solutions.  We can organize this information in a table.

  • Choose values for one variable

When the equation is written in “y=” format it is easier to choose values for x and then find the y’s.  Although you are allowed to choose any x values, for this table I have used a standard set of x’s so that we have a variety.  Some negatives, some positives and zeros.

xy
-2
-1
0
1
2
  • Find the values for the other variable

To find the y’s substitute the value of x into the equation and simplify to find y.

xy
-22(-2)-5=-4-5=-9
-12(-1)-5=-2-5=-7
02(0)-5=0-5=-5
12(1)-5=2-5=-3
22(2)-5=4-5=-1

2.  Plot the ordered pair solutions

Using the table above we have 5 ordered pairs (-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)

Plot the ordered pairs using the rectangular coordinate system.

3.  Draw the line or curve that connects the ordered pairs.

When you look at the plotted ordered pairs you should see a pattern the that points make.  In this case, the points form a straight line.

The line represents all of the ordered pairs that are solutions to the equation y= 2x-5.

Here is a youtube video with a similar example