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Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Unit 5: Polynomial Functions
Section 5.1: Characteristics and Graphs of Polynomials
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Polynomial function a function that contains only the operations of multiplication and additions/subtraction, with real number coefficients, whole number exponents and two variables x and f(x) or y.
Examples of Polynomials NonExamples of Polynomials
Vocabulary
n must be 0 or positive integer...no fractions, etc.
Degree the value of the highest exponent a polynomial function.
Leading Coefficient the coefficient of the term with the greatest degree (exponent) on a polynomial function.
Constant term the term in the polynomial function with no variable.
Degree: 3
Leading coefficient: 7
Constant term: 4
Polynomial functions are named according to their degree and their degree determines the shape of the function.
Polynomial functions of degrees 0, 1, 2, and 3 are called constant, linear, quadratic, and cubic functions, respectively.
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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You Try!
Determine whether each of the following is a polynomial function. If it is, state
the leading coefficient, degree and constant term.
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Domain the set of all possible xvalues which will make the function "work" and will output real yvalues. Range the complete set of all possible resulting yvalues of the dependent variable.
End Behaviour is the description of the shape of the graph, from left to right, on the coordinate plane. It is the behavior of the yvalues as x becomes large in the positive or negative direction (i.e. as x ∞, and x ∞).
Turning Point any point where the graph of a function changes from increasing (yvalues) to decreasing (yvalues) or from decreasing to increasing.
Terminology
a) This curve has two turning points. b) This curve does not have any!
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Functions you have previously studied
1. Constant Function, f(x) = b (Degree = 0)
Type of Polynomial Degree (n) Number of xintercepts # of turning points
Domain Range End Behaviour
f(x)=3
f(x)=2
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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2. Linear Functions, f(x) = ax + b (Degree = 1)
Type of Polynomial Degree (n) Number of xintercepts # of turning points
Domain Range End Behaviour
f(x)=2x+1
f(x)= 1 x32
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3. Quadratic Functions, f(x) = ax2 + bx + c (Degree = 2)
Type of Polynomial Degree (n) Number of xintercepts # of turning points
Domain Range End Behaviour
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Let’s now explore Cubic Functions!Use graphing technology to sketch the graph of each cubic function and complete the table:
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Investigation Questions:
1. How are the sign of the leading coefficient and the end behavior related?
2. How are the degree of the function and the number of xintercepts related?
3. How is the yintercept related to the equation?
4. How many turning points can a cubic function have?
5. In general, how are the number of turning points and the degree related?
6. What is the domain and range for cubic functions?
7. Explain why quadratic functions have maximum or minimum values, but cubic polynomial functions have only turning points?
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4. Cubic Functions, f(x) = ax3 + bx2 + cx + d (Degree = 3)
Type of Polynomial Degree (n) Number of xintercepts # of turning points
Domain Range End Behaviour
Section 5.1 Exploring Graphs of Polynomial Functionssoln.notebook
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Summary Points for Polynomials of Degree 3 or less:
The graph of a polynomial function is continuous
Degree determines the shape of graph
Degree = max # of xintercepts
There is only one yintercept for every polynomial and it is equal to the constant term
The maximum number of turning points is one less than the degree. That is, a polynomial of degree n, will have a maximum of n – 1 turning points.
The end behavior of a line or curve is the behavior of the yvalues as x becomes large in the positive or negative direction. For linear and cubic functions the end behavior is opposite to the left and right, while it is the same for quadratic functions.
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