**Grade 12 – Advanced Functions – Polynomial Functions**

** ****Power Functions**

- A
**polynomial expression**has the form:

a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-1}+ … + a_{3}x^{3}+ a_{2}x^{2}+ a_{1}x+ a_{0}

- Where:
**Exponent**(n) is a whole number- x is the
**variable** - the
**coefficients**a_{0},a_{1},…,a_{n}are real 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:

f(x) = a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-1}+ … + a_{3}x^{3}+ a_{2}x^{2}+ a_{1}x+ a_{0}

- A
**power function**is a polynomial of the form y=ax^{n}, 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.

- Positive leading coefficient (+)
- 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.

- Positive leading coefficient (+)
- 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 the x-axis.
- An even function satisfies the property f(-x) = f(x) for all x in its domain and is symmetric about the y-axis. An even-degree polynomial function is an even function if the exponent of each term is even.
- An even function satisfies the property f(-x) = -f(x) for all x in its domain and is rotationally symmetric about the origin. An odd-degree polynomial function is an odd function if the exponent of each term is odd.

**Determining Odd or Even Functions**

**Even Functions**

- In even functions, f(x)=f(-x), by algebraically proving it.
- They are always symmetrical along the Y-axis
- Their exponents always add to an even number

**Odd functions**

- In odd functions, f(-x) = -f(x), by algebraically proving it.
- Always point symmetry at the origin

** **

**How to graph polynomial functions**

- Determine degree of function (This will allow you to know what type it is)
- Determine and graph the X-Intercepts
- Look at the sign of the coefficient and determine end behavior
- Look at each intercept and determine if it’s a single, double, or triple root.
- Graph the lines with everything above in mind

**Transformations of a polynomial function**

- Translations occur in the form:

y = a[k(x – d)]^{n} + c

**a**- a<0, reflection along the x-axis. Graph opens downwards and creates a maximum point
- a>1, vertical stretch by a factor of a.
- 0<a<1, vertical compression by factor of a.
- a>0, no reflection. Graph opens upwards and creates a minimum point.

**k**- k<0, reflection along y-axis
- 0<k<1, horizontal stretch by a factor of 1/k
- k>1, horizontal compression by factor of 1/k

**c**- C >0, translations C units up
- C <0, translations |c| units down

**d**- d>0, translations d units right
- d<0, translations d units left
- -d is shown as x+d in the equation
- d is shown as x-d in the equation
- Remember to factor out k, if necessary, to obtain the appropriate d value.

**Slopes of Secants and average rate of change**

**Rate of change**is a measure of how quickly one quantity changes with respects to another quantity.- Average rates of change:
- Represent the rate of change over a specified interval
- Correspond to the slope of a secant between two points P
_{1}(x_{1},y_{1}) and P_{2}(x_{2},y_{2}) on a curve. - An average rate of change can be determined by calculating the slope between the two points given using the equation.

**Slopes of tangents and instantaneous rate of change**

- An
**instantaneous rate of change**corresponds to the slope of a tangent to a point on a curve. - An approximate value for an instantaneous rate of change at a point may be determined using:
- A graph, either by estimating the slope of a secant passing through that point or by sketching the tangent and estimating the slope between the tangent point and a second point on the approximate tangent line.
- A table of values by estimating the slope between a point and a nearby point in the table
^{o }An equation, by estimating the slope using very short intervals between the tangent point and a second point found using the equation.