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Lecture

power functions.docx

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Department
Mathematics
Course
MATA30H3
Professor
Ken Butler
Semester
Winter

Description
Power Functions  A polynomial expression has the form: n n-1 n-1 3 2 a n +a xn-1 x + n-2 a x + a x + 3 x+ a 2 1 0  Where:  Exponent (n) is a whole number  x is the variable  the coefficients a ,a0,…1a arenreal numbers  the degree of the function is n, the exponent of the greatest power of x.  a n the coefficient of the greatest power of x, the leading coefficient  a 0 the term without a variable, the constant term.  A polynomial function has the form: n n-1 n-1 3 2 f(x) = a xn+a x n-1x + …n-2a x + a x + a3x+ a 2 1 0 n  A power function is a polynomial of the form y=ax , where n is a whole number. They are often the base of transformations which build on top.  Even-degree power functions have line symmetry in the y-axis.  Odd-degree power functions have point symmetry about the origin. Characteristics of a Polynomial Function  Odd Degree Functions  Positive leading coefficient (+)  Graph end behavior extends from Q3 – Q1.  Negative Leading coefficient (-)  Graph end behavior extends from Q2 – Q4.  Odd degree polynomials have atleast 1 x-intercept, and up to a maximum of n x-intercepts, where n is the degree of the function.  Domain of all odd-degree polynomial functions is {x E R}, and the range is {y E R}.  Odd-Degree Functions have no max/min points.  Odd-Degree functions may have point symmetry.  Even Degree Functions  Positive leading coefficient (+)  Graph end behavior extends from Q2 – Q1.  Will have a minimum point.  Range is {y E R, y <= a}, where a is the minimum value of the function.  Negative Leading coefficient (-)  Graph end behavior extends from Q3 – Q4  Will have a maximum point.  Range is {y E R, y >= a}, where a is the maximum value of the function  Even-Degree polynomials may have zero to a maximum of n-intercepts, where n is the degree of the function.  The domain of all even-degree polynomials is {x E R}.  Even-Degree polynomials may have line symmetry.  For any polynomial function of degree n, the nth differences  Are equal or constant  Have the same sign as the leading coefficient  Are equal to a[n!], where a is the leading coefficient.  Constant Difference = leading coefficient[degree or nth differences !]  D = a [n!] Equations and graphs of a polynomial function  The graph of a polynomial function can be sketched using the x-intercepts, the degree of the function, and the sign of the leading coefficient  The x-intercepts of the graph of a polynomial function are the roots of the corresponding polynomial equation  When a polynomial is in factored form, the zeros can be easily determined from the factors. When a factor is repeated n times, the corresponding zero has order n.  The graph of a polynomial function changes sign only at x-intercepts that correspond to zeros of odd order. At x-intercepts that correspond to zeros of even order, the graph touches but does not cross th
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