Extrema of Polynomials

Extrema of Polynomials

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Extrema

Extrema, or critical points, are maximum or minimum values of a function.

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--> They can usually be spotted by looking for places where the function changes direction, or bends

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--> Using a graph is the BEST strategy for finding extrema​

Subject | Subject

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Multiple Choice

Does this graph have a maximum or minimum value?

• maximum
• minimum
• neither
• both

The highest/lowest point on a specific interval or in a specific graph window.

These should always have numerical or coordinate values.​

​Local/Relative

The highest/ lowest point of the whole function (including off the graph).​

If a function's end goes to infinity, these count as an absolute extrema​

​Global/Absolute

Types of Extrema:

Maximums and Minimums

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Multiple Choice

How many extrema are there in this graph?

• 1

• 2

• 3

• 4

Multiple Choice

How many extrema (maxes and mins) are in the picture?

• 2

• 3

• 4

• 5

Multiple Choice

How would you describe the point at (0,6)?

• Local Minimum

• Absolute Minimum

• Local Maximum

• Absolute Maximum

Multiple Choice

Identify the relative maximum:

• (0, 3)

• (4, 1)

• (3, 0)

• (1, 4)

Multiple Choice

What is the relative maximum value?

• y = 4

• y = 2

• y = 1

• y = -1

Identifying Function Types

You can use the number of extrema to help identify your function type, and vice versa

-->Take the number of extrema and add one. This is the LOWEST possible degree of your function.

OR​

--> Take the degree (highest exponent) and subtract one. This is the HIGHEST number of extrema you could have.​

Subject | Subject

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Multiple Choice

What degree could this function have?

• 3

• 2

• 1

• 0

Multiple Choice

What degree could this function have?

• 2

• 4

• 1

• 0

Multiple Choice

What degree could this function have?

• 5

• 4

• 3

• 2

Multiple Choice

How many relative maximums or minimums might be in this function?

f(x) = 7x⁴ - 6x²

• 3

• 4

• 5

• 6

Multiple Choice

What is the degree of the polynomial shown in the image?

• 1

• 2

• 3

• 0

Multiple Choice

A polynomial of degree n will have at most how many extrema?

• n − 1

• n + 1

• n

• n/2

Multiple Choice

How many relative extrema could the polynomial function  P(x) = 3x⁴+4x-8 have?

• 6

• 5

• 3

• 4

Polynomials SO FAR...

• End Behavior

• Zeroes

• Extrema​

Subject | Subject

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Special Extrema

• Imaginary: the function changes direction WITHOUT CROSSING the x-axis
  

• Multiplicity: the function changes direction EXACTLY on the x-axis (represents 2, 4, or 6 zeroes in the same place)

  ​

Subject | Subject

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Multiple Choice

f(x)=(x+4)(x-3)(x-2)

List the zeros for this function.

• x= -4, x=3, x= -2

• x= -4, x=3, x=2

• x= -4, x=-3, x=2

• x=4, x= -3, x= -2

Multiple Choice

Identify the zeros:

• x=-2, x=-1, x=0

• x=-2, x=0

• x=-1, x=0, x=2

• x=-2, x=0, x=1

Multiple Choice

Which of the following could be the equation of this graph in factored form? (Careful--pay attention to multiplicity.)

• $$f(x)=(x-4)(x-1)^2(x+2)(x+4)$$  

• $$f(x)=(x-4)(x+1)^2(x+2)(x+4)$$  

• $$f(x)=(x-4)(x-1)^2(x-2)(x+4)$$  

• $$f(x)=(x+4)(x+1)^2(x-2)(x-4)$$  

The number next to the highest degree term.​

POSITIVE: points up or trends upward​

NEGATIVE:​points down or trends downward

​Coefficients

EVEN degrees have ends that point in the same direction.

​

ODD degrees have ends that point away from each other​

​Degree

​Using Degrees and Coefficients

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Multiple Choice

Is the leading coefficient positive or negative?

• positive

• negative

Multiple Choice

What is the leading coefficient for the polynomial given by:

y = 3x² + 9x⁵ − 4 + 10x⁸ + 2x⁹ ?

• 2

• 10

• 9

• 3

Multiple Choice

Describe the type of polynomial shown.

• Even degree, negative leading coefficient

• Even Function

• Odd degree, negative leading coefficient

• Even degree, positive leading coefficient

Multiple Choice

Which equation MOST LIKELY matches the graph?

• y = x⁴ - x³ + 3x² + 2

• y = -x⁴ + x³ + 5x + 2

• y = x³ - 2x² + 1x + 3

• y = -x³ - 2x² + 1x + 3

Multiple Choice

What is the degree of the polynomial shown in the image?

• 4

• 5

• 7

• 6

Multiple Choice

What are the extrema of the graph?

• Max = (2, -3), (1, 2)

  Min = (-3, -1), (-1, 4)

• Max = (-3, -1), (-1, 4)

  Min = (2, -3), (1, 2)

• Max = (-1, -3), (4, -1)

  Min = (-3, 2), (2, 1)

• Max = (-3, 2), (2, 1)

  Min = (-1, -3), (4, -1)

Multiple Choice

How many zeroes (real and imaginary) does the polynomial have?

(HINT: check the end behavior)

• 2

• 3

• 4

• 5

Poll

Which BEST describes your mood after this lesson?

• 😀

• 🤠

• 😬

• 😩

• 🤓