Sketching the Graph of a Polynomial, 5

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:

Solution:

Consider the given equation above 

Since the given equation, is already factored, then we can get the x-intercept by setting y = 0. The 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 = (+)⁵ = (+)
If 0 < x < 1, then y = (+)⁵ = (+)
If x > 1, then y = (-)⁵ = (-)

Here's the graph of a polynomial:  

Photo by Math Principles in Everyday Life

Sketching the Graph of a Polynomial, 4

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:

Solution:

Consider the given equation above  

Since the given equation, is already factored, then we can get the x-intercepts by setting y = 0. The x-intercepts are -1 and 3.   

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

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

If x < -1, then y = ¼ (-)³(-) = (+)
If -1 < x < 0, then y =  ¼ (+)³(-) = (-)
If 0 < x < 3, then y = ¼ (+)³(-) = (-)
If x > 3, then y = ¼ (+)³(+) = (+) 

Here's the graph of a polynomial: 

Photo by Math Principles in Everyday Life

Sketching the Graph of a Polynomial, 3

Category: Analytic Geometry

"Published in Vacaville, California, USA

Sketch the graph of a polynomial:

Solution:

Consider the given equation above 

Since the given equation, is already factored, then we can get the x-intercepts by setting y = 0. The x-intercepts are 1 and 3.  

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

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

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

Here's the graph of a polynomial:

Photo by Math Principles in Everyday Life

Sketching the Graph of a Polynomial, 2

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:

Solution:

Consider the given equation above

Since the given equation, is already factored, then we can get the x-intercepts by setting y = 0. The x-intercepts are 1, -1, and 2. 

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

 

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

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

Here's the graph of a polynomial:
 

Photo by Math Principles in Everyday Life

Rotation of a Parabola, 2

Category: Analytic Geometry

"Published in Newark, California, USA"

Given the equation of a parabola

Find the new equation of a parabola if the given parabola is rotated counterclockwise about the origin at. 

Solution:

To illustrate the given problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Since the given angle of rotation is written as inverse tangent function, then we can get the sine and cosine of the given angle of rotation by using basic trigonometric functions of a right triangle. 

 

Photo by Math Principles in Everyday Life

If the given equation is written in rectangular coordinate system, then we need to convert it into polar coordinate system as follows  

  

Next, substitute θ with θ - ϕ and then expand using the sum and difference of two angles formula, we have 

    
Convert the above equation into rectangular coordinate system in order to get its final equation. Therefore, the new equation of a parabola is 

Note: If the axes of any conic sections are not parallel to x and y axes, then the equation of any conic sections has xy term which is the general equation of any conic sections like parabola, ellipse, and hyperbola. Circle has no xy term always in any cases.

Rotation of a Parabola

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Given the equation of a parabola

Find the new equation of a parabola if the given parabola is rotated counterclockwise about the origin at 90°.  

Solution:

To illustrate the given problem, it is better to draw the figure as follows   

Photo by Math Principles in Everyday Life

If the given equation is written in rectangular coordinate system, then we need to convert it into polar coordinate system as follows   

Next, substitute θ with θ - 90° and then expand using the sum and difference of two angles formula, we have  

Convert the above equation into rectangular coordinate system in order to get its final equation. Therefore, the new equation of a parabola is