Sections 9.2 and 9.3 Polynomial Functions

What is a power function? What happens if we add or subtract power functions? A polynomial is a sum (or difference) of power functions whose exponents are nonnegative integers What determines the degree of a polynomial? For example What is the leading term in this polynomial?

Which of the following are polynomials (and what is their degree)?

What are the zeros (or roots) of a polynomial? Where the graph hits the x axis The input(s) that make the polynomial equal to 0 How can we find zeros of a polynomial? For example, what are the zeros of Notice this polynomial is in its factored form It is written as a product of its linear factors

Polynomials Determine the degree and the zeros of the following polynomials? Have students complete this exercise in pairs.

Behavior of Polynomials Using your calculator, graph the following functions and compare the graphs What do you notice about the behavior of the graph at the zeros for m(x) and n(x)? Students should recognize that when there is an even exponent the graph does not go through the x-axis and when there is an odd exponent it does, but it flattens out as it goes through (inflection point)

Behavior of Polynomials x What is the significance of this point? What is the significance of this point? x Students should recognize that when there is an even exponent the graph does not go through the x-axis (and does not change concavity) and when there is an odd exponent it does, but it flattens out as it goes through (inflection point) What behavior do you notice at the zeros of these functions?

Multiplicity of Roots/Zeros When a polynomial, p, has a repeated linear factor, then it has a multiple root. If the factor (x - k) is repeated an even number of times, the graph does not cross the x-axis at x = k. It ‘bounces’ off. Note that the concavity does not change at x = k If the factor (x - k) is repeated an odd number of times, the graph crosses the x-axis, but flattens out at x = k. Note that we will have an inflection point at x = k Click to reveal lines.

Behavior of Polynomials Consider the function: Complete the tables: x f(x) 2 10 100 x f(x) -2 -10 -100 Click to reveal possible answers.

Behavior of Polynomials Consider the function: Complete the tables: What can you say about f(x) as x  ∞? What can you say about f(x) as x  -∞? x f(x) 2 8 10 1,000 100 1,000,000 x f(x) -2 -8 -10 -1,000 -100 -1,000,000 Click to reveal possible answers.

Limit Notation: Another way to notate long-run or end-behavior of functions is by using “limit notation.” We can notate “the limit of f(x) as x goes to infinity” by writing: The above expression signals you to evaluate what the output value of the function f approaches as x gets larger and larger. We can notate “the limit of f(x) as x goes to negative infinity” by writing: This will be many students first introduction to limit notation.

End Behavior Consider the following two functions x f(x) g(x) % change 2 10 100 Have students fill in table and record values.

End Behavior Consider the following two functions x f(x) g(x) % change 2 68 16 4.25 325% 10 10,580 10,000 1.058 5.8% 100 10,050,620 10,000,000 1.005062 0.506% Point out that the percent difference for these functions gets very small as x gets larger and larger, thus the other terms do not have as much of an influence

End Behavior Consider the graphs following two functions Let’s see what happens as we zoom out Be sure to point students attention to the fact that locally these are very different graphs.

End Behavior Consider the graphs following two functions Let’s see what happens as we zoom out some more Be sure to point students attention to the fact that locally these are very different graphs.

End Behavior Consider the graphs following two functions Let’s see what happens as we zoom out some more Be sure to point students attention to the fact that locally these are very different graphs, but the end behavior is very similar for both graphs.

End Behavior Consider the graphs following two functions Be sure to point students attention to the fact that locally these are very different graphs but that the end behavior is almost identical for both graphs.

End Behavior Consider the following two functions Find the following: A functions end behavior is determined by its leading term Allow students to calculate 4 limits. If notion that end behavior does not emerge from student discussion, ask questions that will lead students to that conclusion

End Behavior Both end behavior and degree are determined by the lead term Is there any relationship between the degree of a polynomial and its end behavior? Students should recognize the relationship between even/odd degree and end behavior of polynomials. The graphs can be used to give them a hint.

Find a possible polynomial for the following graph Is it the only possibility? What is the minimum possible degree? There are multiple answers possible for this graph depending on how many times a student assumes a zero is repeated. The graph was created using a triple root at -1 and a double root at 3. The graph also goes through the point (0,1) if a unique solution is desired.