What are they?

 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.