Graphing Quadratic Equations

1
Graphing Quadratic Equations

• An Interactive PowerPoint Presentation
• by Sam Hunter and Eric Ying

2
What are quadratic equations?

• Quadratic equations are polynomial equations with
  a degree of two.
• The standard form of a quadratic equation is
• yax²bxc

3
Solving Quadratic Equations

• There are many ways to solve quadratic equations
  the most simple and effective method is making a
  table of values.

4
Table Of Values.

• Make a table with columns for all of your
  variables.
• Like the table to the right. ?

X Equation Y







5
Practicing Table of values
A. 45, 22,6,5,-7
X Y3X7 Y
-7 Y3(-7)7
-1 Y3(-1)7
0 Y3(0)7
3 Y3(3)7
8 Y3(8)7
B. -14, 4, 7, 16, 31
C. 14, -4, -7, -16, -31
D. 0, 18, 21, 30, 45
Find the rest of the Y values.
6
Graphing Using your points

• Now we move on to using the points you found with
  your table of values and plotting them on a graph.

7
How to plot points

• Your X value is the number of spaces left or
  right you move from the origin on the X axis,
  your Y value is how far you move up or down on
  your Y axis.

X Y3X7 Y
1 Y3(1)7 10
(1,10)
8
Practice Graphing

• Here is a table with three points.

A
X value Equation Yvalue
1 2(x)²-1(x)2 3
2 2(x)²-1(x)2 8
-1 2(x)²-1(x)2 5
Choose the correct graph for the three points.
C
B
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Origin

• The point (0,0) on a graph where the X and Y
  axis intersect.

(0,0)
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X axis

• The x axis is the line running left and right
  from the origin.

X Axis
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Y Axis

• The Y axis is the line perpendicular to the X
  axis that intersects at the origin.

Y axis
12
WRONG

• Sorry but A is absurd the values are no where
  near correct please click on Travis Pastrana 199
  to go back and try again

13
CORRECT

• Good job your calculations are correct. Click
  Travis Pastrana 199 to proceed to the next part
  of Graphing Quadratic Equations.

14
WRONG

• Your only problem is you have your signs mixed up
  so please go back and try again by clicking on
  Travis Pastrana 199.

http//
15
WRONG

• Your problem is you added the numbers in
  parenthesis' with the 7 throwing your math for a
  loop. To try again please click Travis Pastrana
  199.

16
Correct

• Great job graphing! Click Travis Pastrana to
  proceed on.

17
Wrong

• Im sorry but your graphing needs a little but
  Travis Pastrana 199 will bring you back to the
  question so you may try again.

18
Wrong

• Your graphing needs some work but if you click
  on My good buddy Travis Pastrana here hell help
  you get back to the question so you can try again.

19
The difficulties of graphing quadratics.

• The reason graphing quadratic equations is so
  much harder than graphing linear equations I
  because quadratics have vertices which is a point
  where the line bends . This means you must find
  many more points than you would for a linear
  equation to graph the quadratic correctly.

20
Finding Vertices.

• One way to find the vertex of a quadratic
  equation is to convert the standard form
  (ax²bxcy) into the equation ya(x-h)²k and
  the vertex will be (h,k).
• Example 3x² 2x7y when x2
• 3(2-2)²7
• So the vertex is (2,7)

21
X and Y intercepts

• To find the X and Y intercepts for an equation
  set X to zero ( for the Y intercept) and Y to
  zero (for the X intercept).

22
Interception!

• Fin the X and Y intercepts for the equation
• Y 1x²-5x6.
• A.X intercept 14, Y intercept 31
• B.X intercept 2 or 3, Y intercept6
• C.X intercept 6, Y intercept 10
• D. X intercept 3 or 4, Y intercept 6

23
Wrong!

• Your calculations were incorrect please click
  Travis Pastrana 199 to bring you back to the
  question to try again.

24
GREAT JOB

• Great job! This was a tough one because it had
  two values for the x intercept so you had to find
  both unlike linear equations where you only need
  one.

25
Wrong!

• You may need some more practice on this aspect of
  graphing quadratics but dont worry about it its
  no big deal. Travis Pastrana will take you back
  to the question so you may try again. Just click
  his 199.

26
Wrong!

• You need some more practice on this. Click on
  Travis to go back and try again.

27
Determining slopes

• The slope of a line can be found using the model
  
• This means that the second value of Y minus the
  first value of Y over, the second value of X
  over minus the first value of X is the value of
  the slope.

28
Slopes continued

• The problem with this logic it that you must find
  the slope with points from each side of the
  vertex because there are two slopes the slope of
  the line when its increasing and the slope when
  its decreasing.

29
Practicing slopes

• Now to practice.
• There are 4 points in the table and the vertex of
  the equation is (3,4). Find the slopes for each
  side of the vertex.

A. 4/5 and -4/5
B. 1 and -1
C. 4/3 and -4/3
D. 5/9 and -2/7
5 6
8 9
-2 5
-4 7
30
Correct!

• Smiley is proud!
• Please click Travis ( not the smiley) to continue
  on

31
Sorry but youve made a miss-calculation

• Your math was a wee bit off so frowner the downer
  is sad
• Please click Travis
• (not frowner the downer)
• to continue on.

32
Wrong

• Im sorry but that is incorrect please click Trav
  (not frowner the downer) to return to the
  question to try again.

33
Wrong!

• Im sorry but the slope of you mouth will be
  turning down because you are incorrect. Please
  click Travis ( not frowner the downer) to return
  to the question.

34
Final Graphs

• So now is when you put all of your new quadratic
  solving and graphing skills together to
  completely graph a quadratic equation.
• Try this equation and check your work by
  following Travis to the next few slides.
• 3x²-6x5y

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Table of Values for 3x²-6x5Y
X 3x²-6x5Y Y
0 3(0)²- 6(0) 5Y 5
1 3(1)²-6(1) 5Y 2
6 3(6)²-6(6) 5Y 67
-1 3(-1)²- 6(-1) 5Y 14
-2 3(-2)²- 6(-2) 5Y 29
7 3(7)²- 6(7) 5Y 110
1 or 2 3(x)²-6(x) 50 0
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Finding the vertex and slopes

• Vertex Y 3(x-h)²6
• h5 so the vertex is (5,6).
• Slopes 7-6 so the slope is 1
• 110-67
  43
• 29-14 so the slope is
  15
• -2-(-1)
  1
• So the slopes are 1 and 15
• 43 1

37
Graph it
And there you have it all those lessons and tests
for this beautiful graph of the quadratic
equation Y 3x²-6x5
38
Works cited

• http//picfolies.free.fr/pics/persos/smiley.png
  for the smiley face on slide 30.
• http//www.fiferis.com/wp-content/uploads/2008/05/
  frown.gif for frowner the downer on slides
  31-33.
• http//images.dailyradar.com/media/uploads/action/
  story_large/2009/07/27/travis_pastrana.jpg for
  the Travis Pastrana picture used throughout the
  slideshow.