Polynomial Functions of Higher Degree CH 2.2 Polynomial Functions of Higher Degree

Review….

What are the two characteristics of polynomial functions? 1) SMOOTH 2) Continuous

What are some patterns? If the highest exponent is even, the graph is similar to the graph of f(x)=x² May or may not have real x- intercept(s) If the highest exponent is odd, the graph is similar to the graph of f(x)=x³ Will have real x-intercept(s)

What is the leading coefficient test? The end behavior of a polynomial (aka the way a polynomial eventually rises or falls) can be determined by the polynomial’s degree (highest exponent)

You Try! Determine which functions the following graphs will be similar to x² or x³. Use your graphing calculator to confirm. f(x) = - 𝑥 5 G(x) = 𝑥 4 +1 H(x) = - (𝑥+4) 4

Sort your cards into the 4 boxes as shown

How can we find the zeros of a polynomial function? (A zero is a number x for which f(x) = 0 Aka “x-intercept” ) A function of degree n has at most n real zeros and n-1 relative extrema

What are repeated zeros?

Sketch the graph of f(x)= 1 4 𝑥 4 −2 𝑥 2 +3 1) Apply leading coefficient test 2) Find the real zeros 3) Determine Multiplicity 4) Sketch Graph Steps: apply leading coefficient test step

You Try! EX: Find all real zeros of f(x) = x³-x²-2x 1) Apply leading coefficient test 2) Find the real zeros 3) Determine Multiplicity 4) Sketch Graph 1) Apply leading coefficient test 2) Find the real zeros 3) Determine Multiplicity 4) Sketch Graph

EX 2: Find a polynomial with the given zeros a) – 1/2 , 3 (Mult2) b) 3, 2, -2 You Try! -5 (mult 3), 0 (mult 2)

If we have two points on a graph as shown where the y values are never equal, then for any number d in between f(a) and f(b), there must be a number c between a and b such that f(c)=d. If you find a value x =a at which the polynomial is positive, and another value x=b at which the polynomial is negative, you can conclude there is at least one real zero between those two values

Practice. Get out your Factoring hw from ___ Practice! Get out your Factoring hw from ___. Now make note of where the “zeros” are and sketch each polynomial appropriately.

Next day Practice Quiz Factor x² - 11x + 24 2) Put in VERTEX form by Completing the Square

Quiz Time!

Extra Credit on Quiz Explain in a complete sentence why the function has at least one real zero. The function has at least one real zero because any function with all real coefficients and an odd leading exponent must cross the x-axis at least one time, yielding at least one real root.