MHF4U - Unit 1 - Polynomial Functions

Polynomial Function - A function in which f(x) is a polynomial in terms of the variable x
A polynomial function of degree n is written as:
- f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₂x²+ a₁x + a₀
- Where a_n is the leading coefficient and a₀is the constant term
- Polynomial functions are defined and continuous on all real numbers
Types of polynomials:
- Constant (degree 0) - f(x) = a
- Linear (degree 1) - f(x) = ax + b, a≠0
- Quadratic (degree 2) - ax² + bx + c, a≠0
- Cubic (degree 3) - ax³ + bx² + cx + d, a≠0
- Quartic (degree 4) - ax⁴ + bx³ + cx² + dx, a≠0
- Quintic (degree 5) - ax⁵ + bx⁴ + cx³ + dx² + e, a≠0

To plot a function based on an equation, factor the equation
- Solve for the zeroes - the degree indicates the most possible zeroes
- Solve for the y-intercept
- Connect all points with a continuous curve, then add arrows
Interval Notation - A way of writing subsets of the real number line
- Closed Interval - An interval that includes its endpoints, indicated with [ ]
- Open Interval - An interval that does not include its endpoints, indicated with ( )
  - Infinity is an open interval
- If there are multiple sets of conditions, use interval notation on each part and unite them with a “u”
Some polynomial functions have symmetry
- Line symmetry - the graph can reflect on a vertical axis without appearing different
- Point symmetry - the graph can rotate 180 degrees around a point without appearing different
By analyzing the exponents of each term, and then verifying, an equation can be classified as even, odd, or neither

Leading Coefficient - The coefficient of the term with the highest exponent
Increasing - The graph rises going from left to right along the x-axis
Decreasing - The graph falls going from left to right along the x-axis
Local Maximum - The point where the function changes from increasing to decreasing
Local Maximum Value - The y-coordinate of the local maximum
Local Minimum - The point where the function changes from decreasing to increasing
Local Minimum Value - The y-coordinate of the local minimum
Zeros - Roots, x-intercepts, solutions
End Behaviour - The behaviour of the y-values as x approaches positive infinity and as x approaches negative infinity
Absolute Maximum - The highest point on a graph
Absolute Minimum - The lowest point on a graph
To find the least possible degree of a graph with single roots, count the number of local maximums and minimums, and add one
- A single root cleanly passes through the x-axis
- A double root will bounce off the x-axis in a parabola shape
- A triple root will linger at the x-axis before changing values
Odd functions have at least one root with a maximum of “n” zeroes
Even functions can have no roots and a maximum of “n” zeroes

Even Function - A function where the exponent of each term is even, each x in the domain satisfies f(-x) = f(x), and has line symmetry about the y-axis
Odd Function - A function where the exponent of each term is odd, each x in the domain satisfies -f(x) = f(-x), and has point symmetry around the origin when rotated 180 degrees
To sketch a polynomial:
- Using the factors to find roots, and according to their orders, what type of root
- Find the y-intercept
- Find the degree of the function (defines end behaviour)
- Find the sign of the leading coefficient (defines end behaviour)
  - To find, multiply the function by “a,” substitute all known values, and isolate “a”

Translations are applied to polynomial functions as y = a[k(x-d)]ⁿ +c
Reflection - Flips a graph over an axis of symmetry
- y = -f(x) - Flips about the x-axis
- y = f(-x) - Flips about the y-axis
Translation - Relocates the graph but doesn’t change its shape or size
- y = f(x) + c - Shifts graph “c” units vertically
- y = f(x - d) - Shifts graph “d” units horizontally
Dilation - The graph gets wider or narrower
- y = af(x) - a>1, Vertical stretch, 0<a<1, Vertical compression
- y = f(kx) - Horizontal stretch/compression by a factor of 1/k

Secant Line - A line that goes through two points on the graph of the function
Slope
- m = ∆y/∆x
- m = (y₂ - y₁)/(x₂ - x₁)
Average Rate of Change - The rate of change that is measured over an interval on a continuous curve. It corresponds to the slope of the secant between the two endpoints of the interval
To calculate the average rate of change
- m_avg = (y₂ - y₁)/(x₂ - x₁), where x is typically a unit of time

Instantaneous Rate of Change - A change that takes place over an instant. It corresponds with one point on the curve.
Tangent Line - A line that only contacts one point on the graph of a function
To estimate the instantaneous rate of change, draw a tangent line and use two points to calculate slope
To estimate the instantaneous rate of change using an equation, use:
- m = (y₂ - y₁)/(x₂ - x₁), where x₂is the instant in time, and x₁is a value 0.01 less than x₂