pre-ap pre-calculus chapter 2, section 3 polynomial functions of higher degree with modeling 2013 -...
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Pre-AP Pre-CalculusChapter 2, Section 3
Polynomial Functions of Higher Degree with Modeling
2013 - 2014
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Polynomial Functions of Higher Degrees & Vocabulary Cubic Functions –
Quartic Functions –
Term –
Coefficient –
Leading term –
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Graphical Transformations of Monomial Functions
Describe how the monomial was transferred into the given equation.
𝑔 (𝑥 )=4 (𝑥+1)3 h (𝑥 )=−(𝑥−2)4+5
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Graph the polynomial function, locate its extrema and zeros.
𝑓 (𝑥 )=𝑥3+𝑥
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Graph the polynomial function, locate its extrema and zeros.
𝑓 (𝑥 )=𝑥3−𝑥
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Cubic Functions: 2 with positive leading coefficients, 2 with negative leading coefficients
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Quartic Functions: 2 with positive leading coefficients, 2 with negative leading coefficients
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Theorem: Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most
n – 1 local extrema and at most n zeros.
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According to the theorem, how many zeros and local extrema could the following functions have?
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End Behavior Exploration On the following slides, graph each equation
one at a time. Use the window [-5, 5] by [-15, 15]. Describe the end behavior using and
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𝑓 (𝑥 )=2𝑥3 𝑓 (𝑥 )=−𝑥3
𝑓 (𝑥 )=𝑥5 𝑓 (𝑥 )=−0.5 𝑥7
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𝑓 (𝑥 )=−3 𝑥4 𝑓 (𝑥 )=0.6 𝑥4
𝑓 (𝑥 )=2𝑥6 𝑓 (𝑥 )=−0.5 𝑥2
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𝑓 (𝑥 )=−2 𝑥2 𝑓 (𝑥 )=−0.3 𝑥5
𝑓 (𝑥 )=3 𝑥4 𝑓 (𝑥 )=2.5 𝑥3
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Comparing Graphs
Sketch the graph showing both functions
Zoom out till the graphs look nearly identical.
Note the final window
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Applying Polynomial Theory Graph the
polynomial in a window showing its extrema, zeros, and end behavior.
Describe the end behavior using limits.
𝑓 (𝑥 )=𝑥3+2𝑥2−11𝑥−12
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Applying Polynomial Theory
Graph the polynomial in a window showing its extrema, zeros, and end behavior.
Describe the end behavior using limits.
g
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Find the zeros of the function(algebraically)
𝑓 (𝑥 )=𝑥3−𝑥2−6𝑥
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Find the zeros of the function(algebraically)
𝑓 (𝑥 )=3 𝑥2+4 𝑥−4
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Factors & Multiplicity When a factor is repeated, as in , you can say
the polynomial has a repeated zero. The given function has two repeated zeros: x
= _____ and x = ______. Because the factor (x – 2) occurs three times,
the multiplicity of the zero of the function is 3. (it occurs 3 times)
Because the factor (x + 1) occurs twice, the multiplicity of the zero of the function is 2. (it occurs twice)
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Factors & Multiplicity State the degree and list the zeros of the
polynomial. State the multiplicity of each zero.
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Use the Zoom!! Find all real zeros of
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Dixie Packaging Company has contracted to make boxes with a volume of approximately 484 in3 . Squares are to be cut from the corners of a 20-in. by 25-in. piece of cardboard, and the flaps folded up to make an open box. What size squares should be cut from the corners?
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Factor the given equation
6 𝑥3−22𝑥2+12𝑥
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Stopping Distance Draw a scatter plot of the
data. Find the quadratic
regression model. Superimpose the
regression curve on the graph.
Use the regression model to predict the stopping distance for a vehicle traveling at 25 mpg.
Use the regression model to predict the speed of a car if the stopping distance is 300 ft.
Highway Safety Division
Speed (mph) Stopping Distance (ft)
10 15.1
20 39.9
30 75.2
40 120.5
50 175.9
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Ch. 2.3 Homework Pg. 209 - 212: #’s 4, 9, 10,14, 17,
20, 23, 28, 29, 36, 41 (ignore directions about stating whether it crosses the x-axis at
corresponding x-axis), 53, 67, 73
(14 total problems)
Gray Book: pages 193 - 195