ALGEBRA 2 POLYNOMIAL FUNCTIONS CCS MATH 2015!

DEFINITION OF POLYNOMIAL FUNCTION Definition of Polynomial Functions A polynomial function has the form, where   are real numbers and n is a nonnegative integer. [Source: https://www.chegg.com/homework-help/definitions/polynomial-functions-27]

Howarth’s paraphrase of polynomial function. . . Exponents are all positive, whole numbers First term is highest degree in the polynomial We write it in descending order of terms

Name Calling. . . We classify polynomials based on two things; “Please turn to page 307 in your math book and fasten your peepers on the gold chart in the middle of that page while I sleep!” Name Calling. . . We classify polynomials based on two things; 1. Number of terms 2. Degree of exponent 1 term = Monomial 2 terms = Binomial 3 terms = Trinomial General name is a ‘polynomial’ - the prefix ‘poly’ meaning “many”

Polynomials are ALWAYS continuous! CONTINUOUS FUNCTIONS These are functions that do not have vertical asymptotes, holes, jumps; in them Simply put, you can sketch the function without lifting your pencil from the paper at any point. Linear, Polynomial, Root functions are always continuous functions Rational, Piecewise are generally NOT continous functions Polynomials are ALWAYS continuous!

Increasing/Decreasing Functions A Local Max exists A function is increasing if the following holds true; Increasing Decreasing

Increasing/Decreasing Functions A function is increasing if the following holds true;

Our Calculator can find them for us!! A local min exists here Decreasing Increasing

Relative Extrema These are the points on a function that relative to the rest of the function are the high or low points of the function. If we have a relatively high point, we call it a “relative maxima” or relative max, or local max, or just max. If we have a relative low point, we call it a “relative minima” or relative min, local min, or min for short.

If a function is increasing or decreasing along the entire function, Then we say it is “strictly monotonic.” If a function ONLY INCREASES or ONLY DECREASES on some interval, we would say it is monotonic on the interval. If we consider a function for direct variation such that y varies directly with x, using a constant of proportionality of 2; then on closed interval [0, 4] is the function monotonic? Would you say that the function is globally “strictly monotonic” ?

Finding Zero’s of a Polynomial Function Step 1: Factor if we need to in order to solve and find the zeros as we have been doing Example: “Multiplicity”

“DON’T BE A ZERO! FIND THE FUNCTION!” Example: Find the polyomial function given the zero’s. Also state the multiplicity of each root, and the degree of the function. x = 3, 2, 3, -4 Example 2: x = -1/2, 4

Assignment Page 309 #7 – 12 Page 310 #40 – 48 EVENS

We can find them by hand! Let’s put some ideas together that you already know, to synthesize a new idea that you don’t know yet. . . Idea 1: We know that the slope of a horizontal line is zero Idea 2: We know that a tangent line touches a line at only one point. Idea 3: We know that the slope of the tangent line at the relative extrema’s must be zero! Idea 4: If we could find an equation for that tangent line to our function, set it equal to zero, we could find the coordinates of our relative extremas! CLICK ME!

We have introduced to you the concept of a derivative. A derivative is a function that gives the slope of a tangent line to the original function. We can use the derivative to find many things; of interest to us, currently, is using the derivative to find the relative extrema of a function. Here is a helpful animation of what we have been saying.

Let’s watch a quick video that helps reinforce what we’ve been discussing! Thinkwell, Calculus 1, “Slope of a Tangent Line”

The Slope of a tangent line to our function can be found by. . . Finding the Derivative to our function! A derivative is merely another function that has been “derived” from the original function. It is like a “child function” from the Parent function. I will show you the Power Rule for Derivatives to help us here. I will also show you some notation for derivatives that you should know.

The Power Rule for the Derivative

Application of the Power Rule for Derivatives

We can find the slope of a tangent line for many functions!

Here is our picture; Blue is Original function, Red is derivative function So, if we give you a derivative, can you sketch a possible original function knowing what you know now?

One application You may recall that the vertical position function is; The first derivative of s(t) gives the velocity in terms of time t. The second derivative a(t) gives the acceleration of the particle. How fast is a rock traveling at t = 1.5 seconds after being dropped off of a cliff 860 feet from ground level?

Notations for a derivative

We will watch several videos concerning functions “Functions and the Vertical Line Test” “Identifying Functions” “Function Notation and Finding Function Values” “Finding Domain and Range of a Function” “Piecewise Functions” “all function videos. . .