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  4. Advanced Functions
  5. Characteristics of Polynomials

Characteristics of Polynomials

Domain

  • The Domain refers to all the inputs (or \(x\)) values that are possible for a given function. It is also referred to as the replacement set. and is represented algebraically as D
  • The domain of both even and odd polynomials is all real numbers written as \(x \in ℝ\). This means that we can evaluate polynomial functions for any real x-value. There are no domain restrictions (like asymptotes, jumps, holes, or breaks)

Range

  • The Range refers to the set of outputs (or \(y\)) values that are possible for a given function. It is also referred to as the solution set and is represented algebraically as R.
  • The range for odd polynomials is all real numbers written as \(y \in\mathbb{R}\). There are no range restrictions
  • The range for even polynomials is restricted. They have either a minimum (when there is a positive leading coefficient) or a maximum (where there is a negative leading coefficient).

Symmetry

  • Odd degree power functions follow a point symmetry about the origin \((0, 0)\).
  • Even degree power functions follow a line symmetry about the \(y\)-axis at \(x = 0\)
  • Symmetry for odd and even functions can be algebraically summarized as such:

    • Odd: \(f(-x) = -f(x)\)

    Even: \(f(-x) = f(x)\)

  • Not all polynomial functions are symmetry! Use the definitions above to check for even or odd symmetry

End Behaviours

  • The end behaviour for even and odd polynomial functions are dependent on the leading coefficient and are summarized in the table below:

    • Leading Coefficient Even Odd
    Positive \(x\rightarrow \infty, y\rightarrow+\infty\)
    \(x\rightarrow -\infty, y\rightarrow+\infty\)    		 \(x\rightarrow \infty, y\rightarrow+\infty\)
    \(x\rightarrow -\infty, y\rightarrow-\infty\)
    Negative \(x\rightarrow \infty, y\rightarrow-\infty\)
    \(x\rightarrow -\infty, y\rightarrow-\infty\)    		 \(x\rightarrow \infty, y\rightarrow+\infty\)
    \(x\rightarrow -\infty, y\rightarrow+\infty\)

Intercepts

  • \(y\)-intercepts represent points where the function crosses the \(y\)-axis. Functions can only have \(1\) \(y\)-intercept. They can be determined by setting \(x = 0\) and solving for the corresponding \(y\)-value
  • \(x\)-intercepts represent points where the function crosses the \(x\)-axis. Functions can have multiple \(x\)-intercepts. A polynomial of degree \(n\) can have up to \(n\) \(x\)-intercepts

Turning Points

  • A Turning Point occurs when the function reaches a local maximum or local minimum. Turning points occur when the graph changes direction from either increasing to decreasing or decreasing to increasing.
  • Polynomial functions of degree \(n\) can have up to \(n - 1\) turning point. However there are limitations for even and odd degree polynomials. Even degree polynomials must have an odd number of turning points. Odd degree polynomials must have an even number of turning points.
  • The table below shows the possible number of turning points for a polynomial of different degrees:

    • Degree Possible # of Turning Points
    \(1\) \(0\)
    \(2\) \(1\)
    \(3\) \(2\)
    \(4\) \(1, 3\)
    \(5\) \(2, 4\)
    \(6\) \(1, 3, 5\)

Finite Differences

  • Finite Differences can be used to determine the degree of a polynomial using a table of values
  • For an nth degree polynomial, the nth differences will be constant. For instance a cubic function (or function with degree of \(3\) will have constant third differences
  • They can algebraically expressed as:
    • Finite Differences \(= a_n \cdot n!\)

  • The equation states that the finite differences is equal to the leading coefficient (\(a_n\)) multiplied by the factorial of the degree \(n!\)
Determine the degree of the functions based on the following tables using Finite Differences.
X Y
1 0
2 2
3 6
4 12
5 20
6 30
7 42

In order to determine the degree of the function, we can calculate the differences to identify the pattern.

First Differences(ΔY): \(2, 4, 6, 8, 10, 12\)

Since the First Differences aren't constant, we can move on to calculating Second Differences:

Second Differences(Δ²Y): \(2, 2, 2, 2, 2\)

Since the Second Differences are constant, we can determine that the function's degree is \(\boldsymbol{2}\) and is therefore quadratic.

X Y
1 6
2 9
3 20
4 44
5 86
6 151
7 244

In order to determine the degree of the function, we can calculate the differences to identify the pattern.

First Differences(ΔY): \(3, 11, 24, 42, 65, 93\)

Since the First Differences aren't constant, we can move on to calculating Second Differences:

Second Differences(Δ²Y): \(8, 13, 18, 23, 28\)

Since the Second Differences aren't constant, we can move on to calculating Third Differences:

Third Differences(Δ³Y): \(5, 5, 5, 5\)

Since the Third Differences are constant, we can determine that the function's degree is \(\boldsymbol{3}\) and is therefore cubic.

Determine the symmetry, domain, range, end behaviours, and potential # of turning points for the following functions:

\(f(x) = -3x^2 + 7\)

The function is even since it has an even leading degree \(2\).

The function's domain is {\(x \in \mathbb{R}\)}.

The function's range is {\(y \in \mathbb{R} | y \geq 7\)}. Given that the function is even and has a negative leading coefficient, this means it has a maximum (\(7\)).

The function's end behaviours can be summarized below:

\(x \rightarrow \infty, y \rightarrow -\infty\)

\(x \rightarrow -\infty, y\rightarrow -\infty\)

The function only has \(\boldsymbol{1}\) potential turning point.

\(g(x) = 6x^9 - 3\)

The function is even since it has an even leading degree \(9\).

The function's domain and range are \(\boldsymbol{\{x \in \mathbb{R}\}}\) and \(\boldsymbol{\{y \in \mathbb{R}\}}\) respectively. Given that its an odd function, there aren't any restrictions for either characteristic.

The function's end behaviours can be summarized below:

\(x \rightarrow \infty, y\rightarrow +\infty\)

\(x \rightarrow -\infty, y\rightarrow -\infty\)

The function has \(\boldsymbol{8}\) potential turning points.

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