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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