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) = xn

If n is even, the graph is similar to the graph of f(x) = x2

The graph touches the x-axis at the x-intercept. If n is odd, the graph is similar to the graph of f(x) = x3

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 = xn and each specified transformation.

Example 2 y = x3

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 = xn and each specified transformation.

Example 2 y = x3

b) f(x) = -2x3

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 = xn and each specified transformation.

Example 2 y = x3

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 xxxf 4)( 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.

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function.

Example 5 45)( 24 xxxf

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 xxxf 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.

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 f2. 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 xxxxf 2)( 23

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 9xxxxf 3612)( 23

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 1024 22)( xxxf

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 14)( 235 xxxxxf

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 3,3,2

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.

Find a polynomial function that has the given zeros.

Example 13 112,112,3

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.

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) = 3x4 – 4x3

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) = 2x3 – 6x2

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 113-114 (51, 53, 57, 59, 61, 65-68all, 69, 73, 75, 83, 85)