Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Polynomial Functions of Higher Degree Part (Section 2-2)

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function.  The graph of a polynomial function must be continuous (ie it has no breaks, holes, or gaps)  You should be able to draw the graph of a polynomial function without lifting your pencil from the paper.  The graph of a polynomial function should only have smooth rounded turns in it, no sharp turns.

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function.

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. When looking at a function that is of the form f(x) = x n If n is even, the graph is similar to the graph of f(x) = x 2 The graph touches the x-axis at the x-intercept. If n is odd, the graph is similar to the graph of f(x) = x 3 The graph crosses the x-axis at the x-intercept

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of y = x n and each specified transformation. Example 2 y = x 3 a) f(x) = (x – 3) 3

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of y = x n and each specified transformation. Example 2 y = x 3 b) f(x) = -2x 3

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of y = x n and each specified transformation. Example 2 y = x 3 c) f(x) = (x – 4) 3 + 1

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function.

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Pg 105

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Example 4

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Example 5

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Example 6

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Example 7

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Extrema- the maximums and minimums of a function. Relative minimum- the least value of a function on an interval. Relative maximum- the greatest value of a function on an interval. Repeated zero- if (x-a) k, k>1 is a factor of a polynomial, then x=a is a repeated zero. Multiplicity- the number of times a zero is repeated. The following are equivalent statements…. 1.x = a is a zero of the function f 2.x = a is a solution of the polynomial equation f(x)=0. 3.(x-a) is a factor of the polynomial f(x) 4.(a, 0) is an x-intercept of the graph of f. For a polynomial of degree n: The function f has at most n zeros. The graph of f has at most n-1 (relative maximums or minimums)

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find all the real zeros of the polynomial function algebraically. Example 8

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Example 9

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Example 10

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Use a graphing utility to graph the function and approximate any real zeros and relative extrema. Example 11

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. HW #24 pg 112 – 113 (1-8all, 9-47 odd)

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. For a polynomial function, a factor of (x-a) k, k > 1, yields a repeated zero at x = a of multiplicity k.  If k is odd, the graph crosses the x-axis at x = a  If k is even, the graph touches the x-axis (but does not cross thex-axis at x = a)

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find a polynomial function that has the given zeros. Example 12

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find a polynomial function that has the given zeros. Example 13

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find a polynomial function with the given zeros, multiplicities and degree. Example 14 Zero: -5, multiplicity 1 Zero: 0, multiplicity 2 Degree: 3 Red =typo in notes

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Find a polynomial function with the given zeros, multiplicities and degree. Example 15 Zero: 0, multiplicity 2 Zero: 2, multiplicity 2 Degree: 4 Red =typo in notes

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of a polynomial function that satisfies the given conditions. Example 16 Third-degree polynomial with two real zeros and a positive leading coefficient.

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of a polynomial function that satisfies the given conditions. Example 17 Fourth-degree polynomial with three real zeros and a negative leading coefficient.

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of the function by (a) applying leading coefficient test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points and (d) drawing a continuous curve through the points. Example 18 f(x) = 3x 4 – 4x 3

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. Sketch the graph of the function by (a) applying leading coefficient test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points and (d) drawing a continuous curve through the points. Example 19 f(x) = 2x 3 – 6x 2

Essential Question: How do you sketch the graphs of polynomial functions? Students will write a summary of how to sketch a graph of a polynomial function. HW #25 pg (51, 53, 57, 59, 61, 65-68all, 69, 73, 75, 83, 85)