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Polynomial Functions

Polynomial Functions. What are they?. Polynomial Functions. Transformations: are the same as transformations for any function y = a f(k(x-d)) + c Reflections occur is either a or k are negative a= vertical stretch by a factor of absolute a k= horizontal stretch by a factor of absolute k

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Polynomial Functions

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  1. Polynomial Functions What are they?

  2. Polynomial Functions • Transformations: are the same as transformations for any function • y = a f(k(x-d)) + c • Reflections occur is either a or k are negative • a= vertical stretch by a factor of absolute a • k= horizontal stretch by a factor of absolute k • d= horizontal translation (shift) • c= vertical translaton (shift)

  3. Polynomial Functions • Are functions in one variable that contain a polynomial expression. Eg. f(x) = 6x³- 3x²+4x-9. • The exponents are natural numbers (1,2,3,…) • The coefficients are real numbers.

  4. Polynomial Functions • Characteristics • The degree of the function is the highest exponent in the function. • The nth finite differences of a polynomial function of degree n are constant. • The domain is the set of real numbers. • The range may be the set of real numbers or it may have a lower bound or an upper bound, but not both.

  5. Polynomial Functions • Characteristics • The graphs of polynomial functions do not have vertical asymptotes. • The graphs of polynomial functions of degree zero are horizontal lines. The shape of other graphs depends on the degree of the function.

  6. Polynomial Functions • Polynomial functions of the same degree have similar characteristics. • The degree and the leading coefficient in the equation of a polynomial function indicate the end behaviours of the graph. • The degree of the polynomial function provides information about the shape, the turning points, and the zeros of the graph.

  7. Polynomial Functions • End Behaviours

  8. Polynomial Functions • Turning Points • A polynomial function of degree “n” has at most “n-1” turning points (changes of direction).

  9. Polynomial Functions • Number of Zeros • A polynomial function of degree “n” may have up to “n” distinct zeros. • A polynomial function of odd degree must have at least one zero. • A polynomial function of even degree may have no zeros.

  10. Polynomial Functions • Symmetry • Some polynomial functions are symmetrical about the y-axis. They are even functions where f(-x) = f(x). • Some polynomial functions are symmetrical about the origin. These functions are odd functions where f(-x)=-f(x). • Most polynomial functions are neither even or odd.

  11. Polynomial Functions • The Equation of a Polynomial Function • The zeros of a polynomial function are the same as the roots of the related polynomial equation, f(x) = 0. • Steps to determining the equation of a polynomial function in factored form: • Substitute the zeros into the general equation of the appropriate family of polynomial functions • i.e. f(x) = a(x-s)(x-t)(x-u)(x-v) etc. • Substitute the coordinates of an additional point for x and y, solve for “a” to determine the equation.

  12. Polynomial Functions • Final Thoughts for Now • If any of the factors of the function are linear, then the graph will “act” like a straight line near this x-intercept. • If any of the factors of the function are squared, then the corresponding x-intercepts are turning points. We say that factor is “order 2”. The x-axis is tangent to the graph at this point. The graph resembles a parabola near these x-intercepts

  13. Polynomial Functions • Final Thoughts for Now • If any of the factors of a polynomial function are cubed, then there will be a point of inflection at these x-intercepts.The graph will have a cubic shape near these x-intercepts.

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