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Characteristics of Polynomial Functions
Transcript of Characteristics of Polynomial Functions
• Any point on the graph where the function switches from increasing to decreasing or vise versa
• Also an x-value where a local maximum or minimum happens
• Never more than the Degree minus 1
Even or Odd Function?
The end behavior of a function is a description of what happens to the value of f(x) as x approaches infinity and negative infinity.
Degree of a Polynomial
The degree of a polynomial function is the highest exponent to which the dependent variable is raised.
Ex: f(x) = 4x5 – x2 + 5 is a fifth-degree polynomial (highest exponent is 5)
The coefficient of the term with the highest degree is called the leading coefficient
The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.
There is no height greater than f(x)
What is the leading coefficient of this function?
- 7x3 + 6x^2 + 2x - 5
The Leading Coefficient of - 7x3 + 6x2 + 2x - 5
is – 7.
Rules for End Behaviors: Even Function
If the leading coefficient of the terms with the greatest exponent both positive and negative infinity, f(x) approaches positive infinity at both ends
If the leading coefficient is negative, f(x) approaches negative infinity at both ends
Rules for End Behaviors: Odd Functions
If leading coefficient is positive, the function increases as x increases and decreases as x decreases
If the leading coefficient is negative, the function decrease as x increases and increases as x decreases.
Roots of a Polynomial Function
The roots of a function are the x values for which the functions equals zero.
To solve for the roots of a function, set the function equal to 0 and solve for x
The easiest way to find the zeros from the standard form is to factor the equation.
The roots in this situation are -4 and 3 since those are the values at which the function equals 0.
f(x) = f(–x).
Even functions are symmetrical with respect to the y-axis.
Equation: f(-x) = -f(x).
Odd functions are symmetrical with respect to the origin.
f(x)=-(x)^3 + x
f(-x)=-(-x)^3 = (-x)
=-(x^3) – x
-f(x)=-(-x^3 + x)
Opposite to maximum
There is no point lower then f(x)
The maximum or minimum over the whole function
There is only one absolute maximum/minimum, but there can be more than one local maximum or minimum
In some cases there are no turning points