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)
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.
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.
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.
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.
End Behaviours DegreeLeading Coefficient Starts inEnds in Oddpositive3 rd quadrant1 st quadrant Oddnegative2 nd quadrant4 th quadrant Evenpositive2 nd quadrant1 st quadrant Evennegative3 rd quadrant4 th quadrant
Turning Points A polynomial function of degree “n” has at most “n-1” turning points (changes of direction).
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.
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.
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.
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
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.