new day 5 examples
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Characteristics of Polynomial Functions
Recall, standard form of a polynomial:
+ ... + a
x is a variable the coefficients a are real numbers
of a polynomial is the highest power of the variable in the equation.
leading coeffcient is the coeffcient of the
- Linear (degree 1)- Quadratic (degree 2)
(degree 3)- Quartic (degree 4)- Quintic (degree 5)
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The graph of a polynomial is smooth and continuous- no sharp corners and can be drawn without lifting a pencil off a piece of paper
Polynomials can be described by their degree:- Odd-degree polynomials (1, 3, 5, etc.)- Even-degree polynomials (2, 4, etc.)
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Positive, oddfalls to the left at -rises to the right at +∞
End behaviour:
Negative, oddrises to the left at -falls to the right at +∞
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Positive, evenrises to the left at -rises to the right at +∞
End behaviour:
Negative, evenfalls to the left at -falls to the right at +∞
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Can also have a degree of 0..
Constant function
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A point where the graph changes from increasing to decreasing is called a local maximum point
A point where the graph changes from decreasing to creasing is called a local minimum point
A graph of a polynomial function of degree n can have at most n x-intercepts and at most (n - 1) local maximum or minimum points
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Polynomial Matching
What to look for?- degree- leading coefficient- even or odd- number of x-intercepts- number of local max/min- end behaviour
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Graphing Polynomials
of any polynomial fuction y = f(x) correspond to the x-intercepts of the graph and the roots of the equation, f(x) = 0.
ex. f(x) = (x - 1)(x - 1)(x + 2)
If a polynomial has a factor that is repeated then x = a is a zero of multiplicity
x = 1 (zero of even multiplicity)
x = -2 (zero of odd multiplicity)
sign of graph changes
sign of graph does not change
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