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Showing posts with label Analytic Geometry. Show all posts
Showing posts with label Analytic Geometry. Show all posts

Saturday, November 29, 2014

Sketching the Graph of a Polynomial, 9

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:


Solution:

Consider the given equation above  


Did you notice that the trinomial inside the parenthesis is factorable by the product of two binomials? Let's factor the above equation as follows





If we set y = 0, then the x-intercepts of the given equation are 3, and -1.
    
If we set x = 0, then the y-intercept of the given equation is 





Since we now the x-intercept, then we can sketch the location or direction of a curve as follows  

If x < -1, then y = (-)²(-)² = (+)
If -1 < x < 0, then y = (-)²(+)² = (+)
If 0 < x < 3, then y = (-)²(+)² = (+)
If x > 3, then y = (+)²(+)² = (+)

Here's the graph of a polynomial:   

Photo by Math Principles in Everyday Life
 

Friday, November 28, 2014

Sketching the Graph of a Polynomial, 8

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:


Solution:

Consider the given equation above 


Did you notice that the given equation can be factored by grouping? Let's factor the above equation as follows  





and then it is factorable by the difference of two cubes as follows  




If we set y = 0, then we can solve for the value of x at each factors which are the x-intercepts.

For , the value of x is





For , the value of x is


 






Since the roots of the second factor are imaginary numbers, then we cannot accept those values. Because of this, the only x-intercept is 2.
   
If we set x = 0, the y-intercept of the given equation is   





Since we now the x-intercept, then we can sketch the location or direction of a curve as follows    

If x < 0, then y = (-)²(+) = (+)
If 0 < x < 2, then y = (-)²(+) = (+)
If x > 2, then y = (+)²(+) = (+)

Here's the graph of a polynomial:   

Photo by Math Principles in Everyday Life
 
 

Thursday, November 27, 2014

Sketching the Graph of a Polynomial, 7

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:


Solution:

Consider the given equation above   


Did you notice that the given equation has a common factor which is x⁵? Let's factor the above equation as follows 



and then it is factorable by the difference of two squares as follows 



If we set y = 0, then the x-intercepts of the given equation are 0, -3, and 3.

If we set x = 0, then the y-intercept of the given equation is
 
 
 

Since we now the x-intercept, then we can sketch the location or direction of a curve as follows 

If x < -3, then y = (-)³(-)(-) = (-)
If -3 < x < 0, then y = (-)³(+)(-) = (+)
If 0 < x < 3, then y = (+)³(+)(-) = (-)
If x > 3, then y = (+)³(+)(+) = (+)

Here's the graph of a polynomial:    

Photo by Math Principles in Everyday Life
 

Wednesday, November 26, 2014

Sketching the Graph of a Polynomial, 6

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:


Solution:

Consider the given equation above  


Did you notice that the given equation is a perfect trinomial square? Let's factor the above equation as follows



and then it is factorable by the difference of two cubes as follows





If we set y = 0, then we can solve for the value of x at each factors which are the x-intercepts.

For , the value of x is





For , the value of x is 










Since the roots of the second factor are imaginary numbers, then we cannot accept those values. Because of this, the only x-intercept is 1.

If we set x = 0, the y-intercept of the given equation is   




Since we now the x-intercept, then we can sketch the location or direction of a curve as follows   

If x < 0, then y = (-)²(+)² = (+)
If 0 < x < 1, then y = (-)²(+)² = (+)
If x > 1, then y = (+)²(+)² = (+)

Here's the graph of a polynomial:  

Photo by Math Principles in Everyday Life