section 3.1 polynomial functions and models polynomial functions and models
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- SECTION 3.1 POLYNOMIAL FUNCTIONS AND MODELS POLYNOMIAL FUNCTIONS AND MODELS
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- POLYNOMIAL FUNCTIONS A polynomial is a function of the form f(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0 where a n, a n-1,..., a 1, a 0 are real numbers and n is a nonnegative integer. The domain consists of all real numbers.
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- POLYNOMIAL FUNCTIONS Which of the following are polynomial functions?
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- POLYNOMIAL FUNCTIONS SEE TABLE 1
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- POLYNOMIAL FUNCTIONS The graph of every polynomial function is smooth and continuous: no sharp corners and no gaps or holes.
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- POLYNOMIAL FUNCTIONS When a polynomial function is factored completely, it is easy to solve the equation f(x) = 0 and locate the x- intercepts of the graph. Example:f(x) = (x - 1) 2 (x + 3) = 0 The zeros are 1 and - 3
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- POLYNOMIAL FUNCTIONS If f is a polynomial function and r is a real number for which f (r ) = 0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then If r is a (real) zero of f, then (a)r is an x-intercept of the graph of f. (b)(x - r) is a factor of f.
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- POLYNOMIAL FUNCTIONS If (x - r) m is a factor of a polynomial f and (x - r) m+1 is not a factor of f, then r is called a zero of multiplicity m of f. Example: f(x) = (x - 1) 2 (x + 3) = 0 1 is a zero of multiplicity 2. 1 is a zero of multiplicity 2.
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- POLYNOMIAL FUNCTIONS For the polynomial f(x) = 5(x - 2)(x + 3) 2 (x - 1/2) 4 2 is a zero of multiplicity 1 - 3 is a zero of multiplicity 2 1/2 is a zero of multiplicity 4
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- INVESTIGATING THE ROLE OF MULTIPLICITY For the polynomial f(x) = x 2 (x - 2) (a)Find the x- and y-intercepts of the graph. (b)Graph the polynomial on your calculator. (c)For each x-intercept, determinewhether it is of odd or even multiplicity. What happens at an x-intercept of odd multiplicity vs. even multiplicity?
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- EVEN MULTIPLICITY If r is of even multiplicity: The sign of f(x) does not change from one side to the other side of r. The graph touches the x-axis at r.
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- ODD MULTIPLICITY If r is of odd multiplicity: The sign of f(x) changes from one side to the other side of r. The graph crosses the x-axis at r.
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- TURNING POINTS When the graph of a polynomial function changes from a decreasing interval to an increasing interval (or vice versa), the point at the change is called a local minima (or local maxima). We call these points TURNING POINTS.
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- EXAMPLE Look at the graph of f(x) = x 3 - 2x 2 How many turning points do you see? Now graph: y = x 3, y = x 3 - x, y = x 3 + 3x 2 + 4
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- EXAMPLE Now graph: y = x 4, y = x 4 - (4/3)x 3, y = x 4 - 2x 2 How many turning points do you see on these graphs?
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- THEOREM If f is a polynomial function of degree n, then f has at most n - 1 turning points. In fact, the number of turning points is either exactly n - 1or less than this by a multiple of 2.
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- GRAPH: P(x ) = x 2 P 2 (x) = x 3 P(x ) = x 2 P 2 (x) = x 3 P 1 (x) = x 4 P 3 (x) = x 5 P 1 (x) = x 4 P 3 (x) = x 5
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- When n (or the exponent) is even, the graph on both ends goes to When n is odd, the graph goes in opposite directions on each end, one toward + the other toward - .
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- EXAMPLE: Determine the direction the arms of the graph should point. Then, confirm your answer by graphing. Determine the direction the arms of the graph should point. Then, confirm your answer by graphing. f(x) = - 0.01x 7 f(x) = - 0.01x 7
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- EXAMPLE: Graph the functions below in the same plane, first using [- 10,10] by [- 1000, 1000], then using [- 10, 10] by [- 10000, 10000]: Graph the functions below in the same plane, first using [- 10,10] by [- 1000, 1000], then using [- 10, 10] by [- 10000, 10000]: p(x) = x 5 - x 4 - 30x 3 + 80x + 3 p(x) = x 5
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- The behavior of the graph of a polynomial as x gets large is similar to that of the graph of the leading term. The behavior of the graph of a polynomial as x gets large is similar to that of the graph of the leading term.
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- THEOREM For large values of x, either positive or negative, the graph of the polynomial f(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0 resembles the graph of the power function y = a n x n
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- EXAMPLE DO EXAMPLES 9 AND 10
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- CONCLUSION OF SECTION 3.1 CONCLUSION OF SECTION 3.1
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