2.2(b) Notes: Polynomial Functions of Higher Degree Date: 2.2(b) Notes: Polynomial Functions of Higher Degree   Lesson Objective: To graph polynomials and find polynomials given zeros. CCSS: You will need: calculator, colored pens This is Jeopardy!!!: These are the 4 LCTs that determine the end behavior of polynomials.

Lesson 1: Sketching the Graph of a Polynomial 2.2(a) Recap: 1. When n is odd: an > 0 an < 0 Falls left, rises right Rises left, falls right 2. When n is even: Rises left, rises right Falls left, falls right

Lesson 1: Sketching the Graph of a Polynomial 2.2(a) Recap: # of Zeros: A polynomial has at most n real zeros.

Lesson 1: Sketching the Graph of a Polynomial Zeros of f(x) = 2x³ – 8x:

Lesson 1: Sketching the Graph of a Polynomial Zeros of f(x) = 2x³ – 8x: Turning Points: A polynomial graph has at most n – 1 turning points, or relative maxima or minima where the graph changes direction.

Lesson 1: Sketching the Graph of a Polynomial Zeros of f(x) = 2x³ – 8x: Turning Points: A polynomial graph has at most n – 1 turning points, or relative maxima or minima where the graph changes direction. Repeated Zeros: If k is odd, the graph crosses the x-axis. If k is even, the graph touches the x-axis (does NOT cross it).

Lesson 1: Sketching the Graph of a Polynomial Zeros of f(x) = 2x³ – 8x: Polynomial functions can change signs only at its zeros. Between 2 consecutive zeros, a poly-nomial must be entirely positive or entirely negative.

Lesson 1: Sketching the Graph of a Polynomial Zeros of f(x) = 2x³ – 8x: Polynomial functions can change signs only at its zeros. Between 2 consecutive zeros, a poly-nomial must be entirely positive or entirely negative. Test Intervals: Intervals between the zeros used to test the sign

Lesson 1: Sketching the Graph of a Polynomial Tips on Graphing Polynomials: Apply LCT. Find Zeros. Plot a few additional points based on Turning Points.

Lesson 1: Sketching the Graph of a Polynomial A. Sketch the graph of f(x) = 2x³ – 8x. 1. LCT (from 2.2(a) Lesson 2): 2. Zeros (from 2.2(a) Lesson 3): 3. Additional Points: x f(x)

Lesson 1: Sketching the Graph of a Polynomial B. Sketch the graph of f(x) = - 1 4 x⁴ + 3 2 x³ – 9 4 x². 1. LCT: 2. Zeros: 3. Additional Points: x f(x)

Lesson 2: Finding Polynomial Functions Find a polynomial function that has the given zeros. -4, 5

Lesson 2: Finding Polynomial Functions Find a polynomial function that has the given zeros. 0, 2, 5

Lesson 2: Finding Polynomial Functions Find a polynomial function that has the given zeros. -2, -1, 0, 1, 2

Lesson 2: Finding Polynomial Functions Find a polynomial function that has the given zeros. 2, 4 + 5 , 4 – 5

Lesson 3: Finding Polynomial Functions Find a polynomial function of degree n that has the given zeros. Zero(s) Degree x = -3 n = 2

Lesson 3: Finding Polynomial Functions Find a polynomial function of degree n that has the given zeros. Zero(s) Degree x = -8, -4 n = 2

Lesson 3: Finding Polynomial Functions Find a polynomial function of degree n that has the given zeros. Zero(s) Degree x = 9 n = 3

Lesson 3: Finding Polynomial Functions Find a polynomial function of degree n that has the given zeros. Zero(s) Degree x = -4, -1, 3, 6 n = 4

2.2(b): DIGI Yes or No Apply the LCT, find the zeros, and find a few additional points within the test intervals to sketch the graph of Find a polynomial that has 1 + 3 and 1 – 3 as its zeros. Find a polynomial of degree 5 that has x = -3, 1, 5 and 6 as its zeros