Essential Question: How do you sketch the graphs of polynomial functions? Students will write a...
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![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/1.jpg)
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)
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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.
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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.
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/4.jpg)
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
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/5.jpg)
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
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/6.jpg)
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
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/7.jpg)
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
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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.
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/9.jpg)
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
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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
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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
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/12.jpg)
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
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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)(
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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)
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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
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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
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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
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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
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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)
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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)
![Page 21: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/21.jpg)
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
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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
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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
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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
![Page 25: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/25.jpg)
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.
![Page 26: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/26.jpg)
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.
![Page 27: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/27.jpg)
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
![Page 28: 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/5697bf781a28abf838c81b45/html5/thumbnails/28.jpg)
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
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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)