Chapter 3: Algebraic Functions

1. Chapter 3: Algebraic Functions

2. Lesson 3.1: Introduction Polynomials have the property that the sum, difference and product of polynomials always produce another polynomial. In this chapter we will be studying rational functions and algebraic functions. Ex: are both algebraic functions; g(x) is a rational function since it is the quotient of two polynomials.

3. Lesson 3.2: Polynomial Functions

4. A polynomial of degree n, where n is a non-negative integer, has the form are all constants where a ≠ 0. 6 -3 1 17 0 17 17 3 2 -5 10 1 0

5. Recall: We have seen that the graphs of polynomials of degree 0 or 1 are lines and the graph of a polynomial of degree 2 is a parabola. Ex 1: Use learned techniques to graph 1st: we start with the graph of the parent function. 2nd: We shift the graph to the right 2 units. 3rd: We multiply by -3 which will reflect the graph over the x-axis. 4th: Shift the graph 1 unit up. Graph together on board.

6. Ex 2: Sketch the graph of 1st: Notice the graph has 3 x-intercepts 2nd: As x becomes large, is much larger than -4x. So, the graph of f(x) will approach , and its graph will be similar. Graph together on board. Note: The graph has origin symmetry, therefore, it is an odd function. End Behavior of a Graph: The behavior of the y-values of points on the curve for large x-values, as well as x-values that are negative but with large magnitude. Note: the end behavior of any polynomial function of degree n depends solely on the leading coefficient, an. All polynomials of degree n≥1 go to as x goes to .

7. Examples of end behavior p. 128 top Zero of a function: Where f(c)= 0; the x-intercepts. Every polynomial function has the property called continuity. Continuity means the graph has no breaks or interruptions, this is the essence of the Intermediate Value Theorem. Intermediate Value Theorem: if f is continuous on [a, b] and if k is any number between f(a) and f(b), then some number c between (a, b) exists with f(c)= k y Note: The Intermediate Value Theorem tells us that continuous functions do not skip over any values in the range. f(b) k f(a) k = f(c) The graph of a polynomial between successive zeros is either always positive or always negative, meaning lying above or below the x-axis. x a c b

8. Ex 3: Graph Solution: 1st: Find the zeros of the function. Check to see if the terms have a common factor. x = 0, x - 2 = 0, x + 1 = 0 x = 2 x = -1 Zeros = x- intercepts = x = -1, 0, 2 2nd: We must now determine whether the graph between successive zeros lies above or below the x-axis Sign graph~ “ – “ means below the x-axis “+” means above the x-axis x x – 2 x + 1 x(x-2)(x+1) ----------------------0++++++++++++++++ ----------------------------------------0++++++ ------------- 0+++++++++++++++++++++ --------------0+++ 0----------------0++++++ below above below above -1 0 1 2

9. We now know the zeros and on what intervals the graph will lie above and below the x-axis. To determine the End Behavior of the graph we consider f(x) in its original form. f(x) has an odd degree of 3 and a leading coefficient of 1, therefore, the polynomial behaves like the parent function, in large magnitude. Now, with all of our info together we can sketch the graph. y Local maximums and local minimums are called local extrema local maximum x local minimum

10. Think Pair Share: Ex 4: Graph Solution: Zeros: x = 0, 1, 2 x2 x - 2 x – 1 x2(x – 2)(x – 1) ++++++++++++0++++++++++++++++ ------------------------------------0+++++++ -----------------------------0++++++++++++ ++++++++++++0+++ 0----- 0++++++++ above above below above -1 0 1 2 Since the degree is 4 and the leading coefficient is positive it will behave like y = x4