POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors...

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POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus

Transcript of POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors...

Page 1: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

POLYNOMIAL FUNCTIONS Mr. Velazquez

Honors Precalculus

Page 2: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

DEFINITION OF A POLYNOMIAL

Polynomials of degree 1 or higher have graphs that are smooth

and continuous.

• By smooth, we mean that the graphs contain only rounded curves

with no sharp corners, kinks or cusps

• By continuous, we mean that the graphs have no breaks or gaps,

and can be drawn without lifting your pencil

Page 3: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

DEFINITION OF A POLYNOMIAL

Page 4: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

END BEHAVIOR OF A POLYNOMIAL

Page 5: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

END BEHAVIOR OF A POLYNOMIAL

Page 6: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

ZEROS OF POLYNOMIALS

If 𝑓 is a polynomial function, then the values of 𝑥 for which 𝑓 𝑥 = 0 are called the zeros of 𝑓. These values of 𝑥 are also the x-intercepts of the graph of 𝑓, and the roots, or solutions, of the polynomial equation 𝑓 𝑥 = 0.

Page 7: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

EXAMPLE: Find all zeros of 𝑓 𝑥 = 𝑥3 + 4𝑥2 − 3𝑥 − 12

By definition, these are the values of 𝑥 for which 𝑓 𝑥 = 0, so we simply set the function equal to zero, factor it, and set the factors equal to zero to solve for 𝑥:

Therefore, the zeros of 𝑓(𝑥) are:

Page 8: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

EXAMPLES: Find all zeros for the following polynomial functions.

𝑓 𝑥 = 3𝑥2 + 16𝑥 − 12

𝑔 𝑥 = 𝑥3 + 2𝑥2 − 4𝑥 − 8

Page 9: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

MULTIPLICITY OF ROOTS

Page 10: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

MULTIPLICITY OF ROOTS

Page 11: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

CLASSWORK, PART 1 (DO NOW!!)

On a separate sheet of paper (which should be kept for part 2 later), answer

the following question:

Page 12: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

INTERMEDIATE VALUE THEOREM

Since 𝑓(𝑎) is negative and 𝑓 𝑏 is

positive, there must be some value of 𝑥between 𝑎 and 𝑏 for which 𝑓 𝑥 = 0

Page 13: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

INTERMEDIATE VALUE THEOREM

Show that the function 𝑓 𝑥 = 𝑥3 − 2𝑥 − 5 has a real zero

between 𝑥 = 2 and 𝑥 = 3.

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TURNING POINTS OF POLYNOMIALS

In general, a polynomial function of degree 𝑛 can have up to 𝑛 − 1 turning points.

Notice how the function on the left is a degree 5 polynomial and has 4 smooth turning points.

Page 15: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

GRAPHING POLYNOMIAL FUNCTIONS

Page 16: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

GRAPHING POLYNOMIAL FUNCTIONS

Page 17: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

GRAPHING POLYNOMIAL FUNCTIONS

Page 18: POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus · POLYNOMIAL FUNCTIONS Mr. Velazquez Honors Precalculus. DEFINITION OF A POLYNOMIAL Polynomials of degree 1 or higher have graphs

CLASSWORK, PART 2On the same sheet of paper as Part 1, sketch the

graph of the function 𝒇 𝒙 = 𝒙𝟑 − 𝟑𝒙𝟐 , and clearly label all zeros and y-intercepts.

HOMEWORK (for 2nd 9 weeks):

2.3 – Pg. 312, #2-46 (evens)