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5.2 (Day One) Graphing Polynomial Functions Date: ___________
What is a Polynomial Function?
Learning Target C : I can write a polynomial function in standard form and identify its degree and leading coefficient.
We have talked about linear, quadratic, and cubic functions, and all of these are examples of polynomial functions , which are categorized by their degree. A linear function has a degree of 1, a quadratic is of degree 2, and a cubic is a degree 3 polynomial function.
Write each polynomial function in standard form. Then, identify its degree and leading coefficient.
A. ๐(๐ฅ) = ๐ฅ^3 + 4๐ฅ^2 โ ๐ฅ^4 + 1 B. ๐(๐ฅ) = ๐ฅ + 9๐ฅ^3 โ 2๐ฅ + 6๐ฅ^2
Given each function in Intercept form, write it in standard form, and identify the degree and leading coefficient.
A. ๐(๐ฅ) = ๐ฅ^2 (๐ฅ + 1) B. ๐(๐ฅ) = โ3๐ฅ(๐ฅ โ 1)(๐ฅ + 2)^2
Standard form of a Polynomial Function of Degree n
๐(๐ฅ)^ = ๐๐๐ฅ๐^ + ๐๐โ 1 ๐ฅ๐โ^1 + โฏ + ๐ 2 ๐ฅ^2 + ๐ 1 ๐ฅ + ๐ 0
Where ๐๐, ๐๐โ 1 , โฆ , ๐ 2 , ๐ 1 , ๐๐๐ ๐ 0 are real number coefficients
Terms must be in order by their exponents, starting with the highest and ending with the lowest
Degree of a Polynomial: the ___________________ exponent when the polynomial is written in standard form.
Leading Coefficient: the coefficient of the _____________ term when the polynomial is written in standard form.
Intercept Form of a Polynomial Function
Where ๐, ๐ฅ 1 , ๐ฅ 2 , โฆ , ๐๐๐ ๐ฅ๐ are real numbers
The polynomial has degree n , where n is the number of variable factors.
Investigating the End Behavior of Simple Polynomial Functions
Learning Target D: I can determine the end behavior of a polynomial function from its degree and leading coefficient.
Relating End Behavior with Degree
Graph the following functions on a graphing calculator to discover the relationship between the degree of a polynomial with its end behavior.:
๐(๐ฅ) = ๐ฅ, ๐(๐ฅ) = ๐ฅ^2 , ๐(๐ฅ) = ๐ฅ^3 , ๐(๐ฅ) = ๐ฅ^4 , ๐(๐ฅ) = ๐ฅ^5 , ๐๐๐ ๐(๐ฅ) = ๐ฅ^6
Relating End Behavior with Leading Coefficient
Compare the graphs of the following functions with the graphs from the functions above to determine the relationship between the leading coefficient of a polynomial and its end behavior:
๐(๐ฅ) = โ๐ฅ, ๐(๐ฅ) = โ๐ฅ^2 , ๐(๐ฅ) = โ๐ฅ^3 , ๐(๐ฅ) = โ๐ฅ^4 , ๐(๐ฅ) = โ๐ฅ^5 , ๐๐๐ ๐(๐ฅ) = โ๐ฅ^6
Fill in the table with your findings:
Type of Function End Behavior with Positive Leading Coefficient
End Behavior with Negative Leading Coefficient Even Degree
Odd Degree
Given each graph, tell whether the degree of the function is even or odd and identify whether the leading coefficient is positive or negative.
A. B. C.
Degree: Degree: Degree:
LC: LC: LC:
Finding the Zeros of a Polynomial Function
Learning Target E: I can find the zeros of a polynomial function in intercept form.
Recall that the zeros of a function give the graphโs ___________________________. Finding the zeros of a polynomial function is easiest when the function is in intercept form. All we need to do is: ______________________________________________________________________.
Find the x-intercepts and state the degree of each polynomial function. ( x-intercepts are always written as ordered pairs, (x, 0) )
A. f ( x )๏ฝ x ( x ๏ซ 3 )( x ๏ญ 1 ) B. f ( x )๏ฝ ( x ๏ญ 4 )^2 ( x ๏ซ 1 )( x ๏ญ 1 )
C. (^) f ( x )๏ฝ ( x ๏ญ 2 )^2 ( x ๏ญ 6 ) D. ๐(๐ฅ) = ๐ฅ(๐ฅ + 5)^3
5.2 (Day Two) Graphing Polynomial Functions Date: _____________
Investigating the Behavior of the Graph of a Polynomial Function at Its Zero Values
Notice some of the factors in the functions above had exponents other than 1, meaning they occur more than once. The number of times a factor occurs is called its multiplicity.
Letโs see how the multiplicity of a factor affects the behavior of the graph at its related x- intercept! Graph each function on a graphing calculator, and sketch them below. Be sure to accurately plot the x-intercepts.
A. f ( x )๏ฝ x ( x ๏ซ 3 )( x ๏ญ 1 ) B. f ( x )๏ฝ ( x ๏ญ 4 )^2 ( x ๏ซ 1 )( x ๏ญ 1 )
C. (^) f ( x )๏ฝ ( x ๏ญ 2 )^2 ( x ๏ญ 6 ) D. ๐(๐ฅ) = ๐ฅ(๐ฅ + 5)^3
Use what you discovered to fill in the table below:
Sketching the Graph of a Polynomial Function in Intercept Form
Learning Target F: I can use end behavior, x-intercepts, and the y-intercept to graph a polynomial function in intercept form.
Sketch the graph of each polynomial function. Identify the x- and y- intercepts.
A. ๐(๐ฅ) = ๐ฅ(๐ฅ + 2)(๐ฅ โ 3) B. ๐(๐ฅ) = โ(๐ฅ โ 4)(๐ฅ โ 1)(๐ฅ + 1)(๐ฅ + 2)
Behavior at x-intercepts
Goes Straight Through Tangent to x-axis (โBouncesโ) โSquigglesโ Through
C. D.
Identify zeros and multiplicities: Identify zeros and multiplicities:
Degree: Degree:
Leading Coefficient: Leading Coefficient:
Equation: Equation:
