Polynomial Functions Algebra III, Sec. 2.2 Objective Use transformations to sketch graphs. Use the Leading Coefficient Test to determine the end behaviors of graphs. Find and use the real zeros. Use the Intermediate Value Theorem to help locate the real zeros.

Important Vocabulary Continuous – the graph has no breaks, holes, or gaps Repeated Zero – the zeros from recurring factors Test Intervals – intervals used to choose x-values to determine the value of the polynomial fn Intermediate Value Theorem – given two endpoints [a, b] of a continuous fn, there is at least one y value for every x between a and b

Graphs of Polynomial Fns Name two basic features of the graphs of polynomial functions. Continuous Only smooth, rounded turns

Example (on your handout) Will the graph of look more like the graph of or the graph of ? Odd degree  similar to x3

Example 1 Sketch the graph of each function. a) Even degree  similar to x2 Vertical shrink Shift right 8

Example 1 Sketch the graph of each function. b) Even degree  similar to x2 Shift down 5

The Leading Coefficient Test For odd degree polynomials, If the LC is positive ______________________________ If the LC is negative ______________________________ For even degree polynomials,

Example 2 Describe the right-hand and left-hand behavior of the graph of each function. a) Even degree Negative LC So the graph falls to the left and to the right

Example 2 Describe the right-hand and left-hand behavior of the graph of each function. b) Odd degree Positive LC So the graph falls to the left and rises to the right

Example (on your handout) Describe the right-hand and left-hand behavior of the graph of each function.

Zeros of Polynomial Fns On the graph of a polynomial function, turning points are ________________________________________________ ________________________________________________ ________________________________________________ the relative minima or relative maxima… points at which the graph changes between increasing and decreasing…

Zeros of Polynomial Fns Let f be a polynomial function of degree n. The graph of f has, at most, __________ turning points. The function f has , at most, __________ real zeros. n n – 1

Zeros of Polynomial Fns Let f be a polynomial function and let a be a real number. Four equivalent statements about the real zeros of f. x = a is a zero of the function f x = a is a solution of the polynomial equation (x – a) is a factor of the polynomial f(x) (a, 0) is an x-intercept of the graph of f

Example 3 Find all the real zeros of f(x). Then determine the number of turning points of the graph of the function.

If a polynomial function f has a repeated zero x = 3 with multiplicity 4, the graph of f ___________ the x-axis at x = 3. If a polynomial function f has a repeated zero x = 4 with multiplicity 3, the graph of f ___________ the x-axis at x = 4. touches crosses

Example 4 Sketch the graph of

Example (on your handout) Sketch the graph of

Intermediate Value Theorem If (􏰀a, f(􏰀a))􏰁􏰁 and (􏰀b, f(b))􏰁􏰁 are two points on the graph of a polynomial function􏰁, then for any number d between f(􏰀a)􏰁 and f(􏰀b)􏰁 there must be a number c between a and b such that f(􏰀c)=􏰁d.

Intermediate Value Theorem The Intermediate Value Theorem helps you locate the real zeros of a polynomial function. If you can find a value x=a at which a polynomial function is positive, and another value x=b at which it is negative, you can conclude that the function has at least one real zero between these two values.

a real zero must be between 1 & 2 Example 6 Use the Intermediate Value Theorem to approximate the real zero(s) of the function. First, create a table of function values. x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 -(0)5+3(0)3-2(0)+2 1 -(1)5+3(1)3-2(1)+2 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 -(0)5+3(0)3-2(0)+2 1 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 -(0)5+3(0)3-2(0)+2 1 -(1)5+3(1)3-2(1)+2 -(2)5+3(2)3-2(2)+2 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 -(0)5+3(0)3-2(0)+2 1 -(1)5+3(1)3-2(1)+2 -(2)5+3(2)3-2(2)+2 -10 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 -(0)5+3(0)3-2(0)+2 1 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 -(0)5+3(0)3-2(0)+2 1 -(1)5+3(1)3-2(1)+2 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 1 2 x -x5+3x3-2x+2 f(x) -2 -1 1 2 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 2 1 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 -1 1 2 x -x5+3x3-2x+2 f(x) -2 -(-2)5+3(-2)3-2(-2)+2 14 -1 -(-1)5+3(-1)3-2(-1)+2 1 2 sign change a real zero must be between 1 & 2

Example 6 Use the Intermediate Value Theorem to approximate the real zero(s) of the function. Now, divide the interval into tenths & evaluate…

Example 6 Use the Intermediate Value Theorem to approximate the real zero(s) of the function. You can also evaluate the graph…