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Degrees of Polynomials and Numbers of Turning Points

Sixth-Degree Polynomial Possible  Graphs

y=ax6+bx5+cx4+dx3+ex2+fx+g

 

Degrees of Polynomials and Numbers of Turning Points
Function Representation

Function:

f(x) = x6 - 3x5 + 2x4 - x3 + 4x2 - 5x + 6

Degree:

6

Leading Coefficient:

Positive

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → ∞
As x → ∞ , f(x) → ∞

Additional Info:

The graph represents a sixth-degree polynomial, specifically the function f(x)=x6−3x5+2x4−x3+4x2−5x+6 . Unlike typical sixth-degree polynomials that may exhibit up to five turning points due to their complex nature, this graph is notable for displaying just a single visible turning point. This characteristic is significant as it deviates from the more common multiple turning points expected in high-degree polynomial functions. The turning point shown here represents a local minimum, indicating the lowest value the function reaches within the visible range of the graph. This unique feature highlights an intriguing aspect of polynomial behavior, where specific coefficients significantly influence the graph’s shape and turning points.
Degrees of Polynomials and Numbers of Turning Points
Function Representation

Function:

f(x) = − 3 x6 − 4 x5 + x2 − 9 x + 7

Degree:

6

Leading Coefficient:

Negative

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → − ∞
As x → ∞ , f(x) → − ∞

Additional Info:

The graph depicted showcases a sixth-degree polynomial f(x)=−3x6−4x5+x2−9x+7y = -3x^6 – 4x^5 + x^2 – 9x + 7 . This particular polynomial graph features a single, prominently visible turning point, marking a significant peak before the curve descends steeply.
Degrees of Polynomials and Numbers of Turning Points
Function Representation

Function:

f(x)=−x6+5x4−3x3+6x2−2x+8

Degree:

6

Leading Coefficient:

Negative

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → − ∞
As x → ∞ , f(x) → − ∞
The graph you see represents a sixth-degree polynomial given by the equation f(x)=−x6+5x4−3x3+6x2−2x+8y = -x^6 + 5x^4 – 3x^3 + 6x^2 – 2x + 8 . This graph illustrates 3 turning points indicating changes in the direction of the curve, which are common features in higher-degree polynomials.
Degrees of Polynomials and Numbers of Turning Points
Function Representation

Function:

f(x)=−x6+5x4−3x3+6x2−2x+8

Degree:

6

Leading Coefficient:

Negative

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → − ∞
As x → ∞ , f(x) → − ∞
The graph you see represents a sixth-degree polynomial given by the equation f(x)=−x6+5x4−3x3+6x2−2x+8y = -x^6 + 5x^4 – 3x^3 + 6x^2 – 2x + 8 . This graph illustrates 3 turning points indicating changes in the direction of the curve, which are common features in higher-degree polynomials.
Degrees of Polynomials and Numbers of Turning Points
Function Representation

Function:

f(x)=25x6+19x5−x4+3x3−7x2+2x−1

Degree:

6

Leading Coefficient:

Positive

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → ∞
As x → ∞ , f(x) → ∞
This graph displays a sixth-degree polynomial, expressed by the equation

f(x)=25x6+19x5−x4+3x3−7x2+2x−1\frac{2}{5}x^6 + \frac{1}{9}x^5 – x^4 + 3x^3 – 7x^2 + 2x – 1

     
Characterized by 3 turning points, the graph showcases the complex behavior of higher-degree polynomials. Each turning point indicates a local extremum—either a maximum or a minimum—where the slope of the function changes direction. The polynomial’s coefficients dictate these fluctuations, demonstrating how subtle changes in the terms can significantly alter the graph’s topology. This visualization helps underscore the intricacies of polynomial functions and their applications in modeling various real-world phenomena.
Function Representation

Function:

f(x)=x6−9x5+12x2

Degree:

6

Leading Coefficient:

Positive

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → ∞
As x → ∞ , f(x) → ∞
The graph depicts a sixth-degree polynomial function given by the equation f(x)=x6−9x5+12x2y = x^6 – 9x^5 + 12x^2 . This function exhibits 5 turning points, highlighted clearly in the diagram. The polynomial demonstrates complex behavior with changes in direction, showing how higher-degree polynomials can have varied and intricate shapes, influenced strongly by the powers and coefficients of the terms involved. This specific graph illustrates four turning points, showcasing both local maxima and minima, which is characteristic for polynomials of higher degrees that involve significant coefficients on higher powers of xx .
Function Representation

Function:

f(x)=−2x6+9x4−12x2

Degree:

6

Leading Coefficient:

Negative

Possible Number of Turning Points:

1, 3, 5

End Behavior:

As x → −∞ , f(x) → − ∞
As x → ∞ , f(x) → − ∞
This graph depicts a sixth-degree polynomial function, represented by the equation f(x)=−2x6+9x4−12x2y = -2x^6 + 9x^4 – 12x^2 . The curve showcases several turning points, which are points at which the direction of the curve changes. These turning points are critical for understanding the behavior of polynomial functions, as they indicate local maxima and minima — places where the function reaches local highest or lowest values. The distribution and nature of these turning points also give insights into the function’s derivative, as turning points occur where the derivative (the slope of the tangent to the curve) equals zero. This specific graph illustrates how even high-degree polynomials with negative leading coefficients can exhibit multiple ups and downs within a range, shaping the curve with multiple peaks and valleys.
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