Evaluating and Graphing Polynomial Functions 6.2 Evaluating and Graphing Polynomial Functions

It is NOT a function if there are negative exponents or variable as A polynomial function is a function of the form f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0 constant term an  0 an leading coefficient descending order of exponents from left to right. n n – 1 n degree Where an  0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a 0 is the constant term, and n is the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. It is NOT a function if there are negative exponents or variable as exponents

You are already familiar with some types of polynomial functions. Here is a summary of common types of polynomial functions. Degree Type Standard Form f (x) = a Constant f (x) = a1x + a 1 Linear 2 Quadratic f (x) = a 2 x 2 + a 1 x + a 3 Cubic f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a f (x) = a4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 4 Quartic

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = x 2 – 3x4 – 7 1 2 SOLUTION The function is a polynomial function. Its standard form is f (x) = – 3x 4 + x 2 – 7. 1 2 It has degree 4, so it is a quartic function. The leading coefficient is – 3.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = x 3 + 3 x SOLUTION The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = 6x 2 + 2 x –1 + x SOLUTION The function is not a polynomial function because the term 2x –1 has an exponent that is not a whole number.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = – 0.5 x +  x 2 – 2 SOLUTION The function is a polynomial function. Its standard form is f (x) =  x2 – 0.5x – 2. It has degree 2, so it is a quadratic function. The leading coefficient is .

f (x) = x 2 – 3 x 4 – 7 f (x) = x 3 + 3x f (x) = 6x2 + 2 x– 1 + x Identifying Polynomial Functions Polynomial function? f (x) = x 2 – 3 x 4 – 7 1 2 f (x) = x 3 + 3x f (x) = 6x2 + 2 x– 1 + x f (x) = – 0.5x +  x2 – 2

Using Synthetic Substitution One way to evaluate polynomial functions is to use direct substitution. Another way to evaluate a polynomial is to use synthetic substitution. Use synthetic division to evaluate f (x) = 2 x 4 + -8 x 2 + 5 x - 7 when x = 3.

Using Synthetic Substitution SOLUTION 2 x 4 + 0 x 3 + (–8 x 2) + 5 x + (–7) Polynomial in standard form Polynomial in standard form 2 0 –8 5 –7 3 x-value 3 • Coefficients Coefficients 6 18 30 105 2 10 35 6 98 The value of f (3) is the last number you write, In the bottom right-hand corner.

Now use direct substitution: Use synthetic division to evaluate f (x) = 2 x 4 + -8 x 2 + 5 x - 7 when x = 3. You get the same answer either way!

HOMEWORK (DAY 1) 333-334/4-46 evens

If “n” is even, the graph of the polynomial is “U-shaped” meaning it is parabolic (the higher the degree, the more curves the graph will have in it). If “n” is odd, the graph of the polynomial is “snake-like” meaning looks like a snake (the higher the degree, the more curves the graph will have in it).

Let’s talk about the Leading Coefficient Test:

Leading Coefficient Test Degree is odd Degree is even L.C. > 0 Start high, End high L.C. < 0 Start low, End low L.C. > 0 Start low, End high L.C. < 0 Start high, End low L.C. = Leading Coefficient

f(x) = x4 + 2x2 – 3x f(x) = -x5 +3x4 – x f(x) = 2x3 – 3x2 + 5 Determine the left and right behavior of the graph of each polynomial function. f(x) = x4 + 2x2 – 3x f(x) = -x5 +3x4 – x f(x) = 2x3 – 3x2 + 5

Tell me what you know about the equation…

Tell me what you know about the equation… Page 261 #53

Tell me what you know about the equation… Page 261 #54

Tell me what you know about the equation… -x^4+3x^2+4

Fundamental Thm of Algebra Zeros of Polynomial Functions: The graph of f has at most n real zeros The “n” deals with the highest exponent!

How many zeros do these graphs have????

x +  is read as “x approaches positive infinity.” GRAPHING POLYNOMIAL FUNCTIONS The end behavior of a polynomial function’s graph is the behavior of the graph as x approaches infinity (+ ) or negative infinity (– ). The expression x +  is read as “x approaches positive infinity.”

GRAPHING POLYNOMIAL FUNCTIONS END BEHAVIOR

f(x) Graph f (x) = x 3 + x 2 – 4 x – 1. –3 –7 –2 3 –1 1 2 23 SOLUTION Graphing Polynomial Functions Graph f (x) = x 3 + x 2 – 4 x – 1. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. x f(x) –3 –7 –2 3 –1 1 2 23

f (x) Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. –3 –21 –2 –1 1 3 2 –16 Graphing Polynomial Functions Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. x f (x) –3 –21 –2 –1 1 3 2 –16 –105

Assignment Day 2 334-335/50-78 evens