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unique quadratic equation in the form y = ax^2 + bx + c

by Brittni Rivera
(Greeley, CO)











































quadratic equation opens in the same direction and shares one of the x-intercepts

A) Create your own unique quadratic equation

• in the form y = ax^2 + bx + c

• that opens the same direction

• and shares one of the x-intercepts of the graph of y = x^2 + 4x - 12.


B) Determine the following

• Explain whether the graph has a maximum or minimum point.

• Find the vertex and x-intercepts of the graph.


C) Use complete sentences and show all work to receive full credit.

Comments for unique quadratic equation in the form y = ax^2 + bx + c

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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


Answer

Part I

A) Create your own unique quadratic equation

• in the form y = ax² + bx + c

• that opens the same direction

• and shares one of the x-intercepts of the graph of y = x² + 4x - 12.




a graph of the equation y = x² + 4x - 12 is shown below



Graph of the quadratic function:  y = x² + 4x - 12





y = x² + 4x - 12 is the equation for a parabola



General form of a quadratic function






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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


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Part II


to convert the equation for a parabola to a quadratic equation, set y = 0

0 = x² + 4x - 12

x² + 4x - 12 = 0

x² + 4x - 12 = 0 is a quadratic equation of the form:

ax² + bx + c = 0

Standard Format of quadratic equation:


Standard format of a Quadratic Equation




The x-intercepts can be found by solving for x.

Finding the value of x (the x-intercepts) can be determined by factoring the quadratic equation, using the quadratic formula, completing the square, graphing, or using the Indian method.

We are going to compare two methods: (1) graphing, and (2) factoring.

(1) graphing

Since we already have a graph of the quadratic function (y = x² + 4x - 12), we can just read the x-intercepts off the graph:

x-intercepts = {-6, 2}


(2) factoring

x² + 4x - 12 = 0

(x + 6)(x - 2) = 0

x + 6 = 0

x + 6 - 6 = 0 - 6

x + 0 = 0 - 6

x = - 6

x - 2 = 0

x - 2 + 2 = 0 + 2

x + 0 = 0 + 2

x = 2

x-intercepts = {-6, 2}






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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


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Part III


Factoring a Quadratic Equation to find the x-intercepts





Both the results obtained by graphing and the results obtained by factoring are the same.


The NEW EQUATION

We know what the factored form of the original equation is.

y = x² + 4x - 12 = (x + 6)(x - 2)

The factored form shows the two intercepts.

(x + 6) = x-intercept of -6

(x - 2) = x-intercept of 2


x-intercepts = {-6, 2}


Changing ONE of the constants in the factored form will create a unique equation which opens in the same direction as the original equation and shares one of the same x-intercepts.

change the + 6 to another number (x + 6)

OR (Do not change both numbers. Change only one of the numbers.)

change the - 2 to another number (x - 2)





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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


---------------------------------------------




Part IV


To create a new equation which shares one of the x-intercepts with the original equation, change one of the constants




As an example, I'm going to change the - 2 to + 10

The factored form of the new equation is:

y = (x + 6)(x + 10)

When the factored form of the new equation is expanded, the general form of the new equation is:

y = x² + 16x + 60


Create a new equation by changing the -2 to a +10





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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


---------------------------------------------




Part V



Factored form of the new equation to the general form





When both the original equation and the new equation are plotted on the same set of coordinates, it is obvious that both open upward and both share the x-intercept: x = -6.


When both the original equation and the new equation are plotted on the same set of coordinates, it is obvious that both open upward and both share the x-intercept:  x = -6.








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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


---------------------------------------------




Part VI



B) Determine the following

• Explain whether the graph has a maximum or minimum point.

• Find the vertex and x-intercepts of the graph.



the original equation

y = x² + 4x - 12 = (x + 6)(x - 2)

x-intercepts

The factored form shows the two intercepts.

(x + 6) = x-intercept of -6

(x - 2) = x-intercept of 2

x-intercepts = {-6, 2}

vertex

the x coordinate of the vertex is given by the formula:

x = -b/2a

b = 4

a = 1

x = -4/(2*1)

x = -2

the y coordinate of the vertex can be computed by inserting -2 for x in the quadratic function:

y = x² + 4x - 12

y = (-2)² + 4*(-2) - 12

y = 4 - 8 - 12

y = - 16

the coordinates of the vertex are:

vertex = (-2, -16)

The vertex of this parabola is a minimum point.


There is no maximum point.


Vertex of the parabola y = x² + 4x - 12.








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Apr 06, 2014
quadratic equation opens in the same direction and shares one of the x-intercepts
by: Staff


---------------------------------------------




Part VII



the new equation

y = x² + 16x + 60 = (x + 6)(x + 10)


x-intercepts

The factored form shows the two intercepts.

(x + 6) = x-intercept of -6

(x + 10) = x-intercept of -10

x-intercepts = {-10, -6}

vertex

the x coordinate of the vertex is given by the formula:

x = -b/2a

b = 16

a = 1

x = -16/(2*1)

x = -8

the y coordinate of the vertex can be computed by inserting -8 for x in the quadratic function:

y = x² + 16x + 60

y = (-8)² + 16*(-8) + 60

y = 64 - 128 + 60

y = - 4

the coordinates of the vertex are:

vertex = (-8, -4)

The vertex of this parabola is a minimum point.

There is no maximum point.



 Vertex of the parabola y = x² + 16x + 60.







Thanks for writing.

Staff
www.solving-math-problems.com



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