Graphing Quadratic Equations

An Interactive PowerPoint Presentation

by Sam Hunter and Eric Ying

What are quadratic equations?

Quadratic equations are polynomial equations with a degree of two.

The standard form of a quadratic equation is:

y=ax²+bx+c

Solving Quadratic Equations

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

Table Of Values.

Make a table with columns for all of your variables.

• Like the table to the right.

X Equation Y

Practicing Table of values

X Y=3X+7 Y

-7 Y=3(-7)+7

-1 Y=3(-1)+7

0 Y=3(0)+7

3 Y=3(3)+7

8 Y=3(8)+7

Find the rest of the Y values.

A. 45, 22,6,5,-7

B. -14, 4, 7, 16, 31C. 14, -4, -7, -16, -31D. 0, 18, 21, 30, 45

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.

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 Y=3X+7 Y

1 Y=3(1)+7 10

(1,10)

Practice Graphing• Here is a table with

three points. 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

5Choose the correct graph for the three points.

A

B

C

Origin

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

X axis

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

X Axis

Y Axis

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

Y axis

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

CORRECT

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

WRONG

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

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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.

Correct

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

Wrong

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

Wrong

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

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.

Finding Vertices.

• One way to find the vertex of a quadratic equation is to convert the standard form (ax²+bx+c=y) into the equation y=a(x-h)²+k and the vertex will be (h,k).

• Example: 3x² +2x+7=y when x=23(2-2)²+7

So the vertex is (2,7)

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).

Interception!

• Fin the X and Y intercepts for the equation:

Y= 1x²-5x+6.A.X intercept: 14, Y intercept :31B.X intercept: 2 or 3, Y intercept:6C.X intercept: 6, Y intercept: 10D. X intercept: 3 or 4, Y intercept : 6

Wrong!

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

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.

Wrong!

• You may need some more practice on this aspect of graphing quadratics but don’t 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.

Wrong!

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

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.

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.

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.

5 6

8 9

-2 5

-4 7

A. 4/5 and -4/5

B. 1 and -1

C. 4/3 and -4/3D. 5/9 and -2/7

Correct!

Smiley is proud!

Please click Travis ( not the smiley) to continue on

Sorry but you’ve 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.

Wrong

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

Wrong!

• I’m 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.

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.

3x²-6x+5=y

Table of Values for 3x²-6x+5=Y

X 3x²-6x+5=Y Y

0 3(0)²- 6(0) + 5=Y 5

1 3(1)²-6(1) +5=Y 2

6 3(6)²-6(6) + 5=Y 67

-1 3(-1)²- 6(-1) + 5=Y 14

-2 3(-2)²- 6(-2) + 5=Y 29

7 3(7)²- 6(7) +5=Y 110

1 or 2

3(x)²-6(x) +5=0 0

Finding the vertex and slopes

Vertex : Y= 3(x-h)²+6 h=5 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) 1So the slopes are 1 and 15 43 1

Graph itAnd there you have it all those lessons and tests for this beautiful graph of the quadratic equation: Y= 3x²-6x+5

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.