Day 5 examples u5w14
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Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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Day 5 examples u5w14
- 1. 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 coefficient is the coefficient of the
- 2. 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.) - Linear (degree 1) - Quadratic (degree 2) (degree 3) - Quartic (degree 4) - Quintic (degree 5)
- 3. Positive, odd falls to the left at - rises to the right at +∞ End behaviour: Negative, odd rises to the left at - falls to the right at +∞
- 4. Positive, even rises to the left at - rises to the right at +∞ End behaviour: Negative, even falls to the left at - falls to the right at +∞
- 5. Can also have a degree of 0.. Constant function
- 6. 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
- 7. Polynomial Matching What to look for? - degree - leading coefficient - even or odd - number of x-intercepts - number of local max/min - end behaviour
- 11. 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
- 12. ex. f(x) = (x - 1)2(x + 3)2
- 13. ex. g(x) = -(x + 2)3(x - 1)2