4-1 Graphing Polynomial Functions

Polynomials A monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents.  Monomial: 5, 2x, xyh, x2 Not a monomial: : x+2, x/y, x3/2 A polynomial is a monomial or a sum of monomials

Polynomial Functions A polynomial function is a function of the form where , the exponents are all whole numbers, and the coefficients are all real numbers. In standard form, a polynomial function is written in order of descending powers.

Polynomial Graphs quadratic cubic quartic

Examples of polynomials Degree Name Example 0 constant 1 linear 2 quadratic 3 cubic 4 quartic 5 quintic

Polynomials versus not polynomials Is each function a Polynomial Function ? If yes, what is its degree? Function Polynomial Degree Yes 1 Yes 2 No Yes No

Evaluating Polynomial Functions Evaluate when Substitute 3 for x in original eqn. Evaluate powers and multiply. Simplify.

End Behavior The end behavior of a function's graph is the behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞). For the graph of a polynomial function, the end behavior is determined by the function's degree and the sign of the leading coefficient.

End Behavior of Polynomial Functions Degree: odd Leading coefficient: positive Degree: odd Leading coefficient: negative

End Behavior of Polynomial Functions Degree: even Leading coefficient: positive Degree: even Leading coefficient: negative

End Behavior of Polynomial Functions Example: evaluate the end behavior of the function It has a degree of 4 (even) and a leading coefficient of -0.2 (negative). f(x) approaches -∞ as x approaches -∞ and as x approaches +∞.

Graphing Polynomial Functions To graph a polynomial function, first plot points to determine the shape of the graph’s middle portion. Then connect the points with a smooth continuous curve and use what you know about end behavior to sketch the graph. Ex: graph X -2 -1 1 2 F(X) 3 -4 -3 The degree is odd and the leading coefficient is negative. So, f(x) approaches +∞ as x approaches - ∞ and f(x) approaches - ∞ as x approaches + ∞.

Solving a Real Life Problem The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function with t representing time and t=1 corresponding to 2001. a. Use a graphing calculator to graph the function for the interval 1 ≤ t ≤ 10. 

Solving a Real Life Problem The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function with t representing time and t=1 corresponding to 2001. b. What was the average rate of change in the number of electric vehicles in use from 2001 to 2010? (1, 18.57) , (10, 100.11) where t = 1 and t = 10 The average rate of change in the number of electric vehicles in use from 2001 to 2010 is about 9.06 thousand electric vehicles per year.

Solving a Real Life Problem The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function with t representing time and t=1 corresponding to 2001. c. Do you think this model can be used for years before 2001 or after 2010? Explain your reasoning. Because the degree is odd and the leading coefficient is positive, V(t) approaches +∞ as t approaches +∞ and V(t) approaches -∞ as t approaches -∞ . The end behavior indicates that the model has unlimited growth as t increases. While the model may be valid for a few years after 2010, in the long run unlimited growth is not reasonable. Notice that in 2000 that v(0) = -1.39. Because negative values of V(t) do not make sense given the context (electric vehicles in use), the model should not be used for years before 2001