Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
A comprehensive guide to understanding and analyzing graphs of polynomial functions. It covers key concepts such as finding zeros using factoring, identifying multiplicities of zeros, interpreting turning points, and applying the intermediate value theorem. Well-structured, includes illustrative examples, and provides practical exercises for reinforcement.
Typology: Study notes
1 / 15
This page cannot be seen from the preview
Don't miss anything!
Using Factoring to Find Zeros of Polynomial Functions Given a polynomial function f, find the x-intercepts by factoring.
EX. Find the x-intercepts of f (x) = x 6 − 3x 4
EX. Find the x-intercepts of h(x) = x 3
Try It # Find the y- and x-intercepts of the function f (x) = x^4 − 19x^2 + 30x.
If a polynomial contains a factor of the form (x − h) p , the behavior near the x-intercept h is determined by the power p. We say that x = h is a zero of multiplicity p. The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. The graph will cross the x-axis at zeros with odd multiplicities. The sum of the multiplicities is the degree of the polynomial function.
EX. Use the graph of the function of degree 6 in Figure 9 to identify the zeros of the function and their possible multiplicities. The first zero occurs at x = −3. The graph touches the x-axis, so the multiplicity of the zero must be even. The zero of −3 has multiplicity 2. The next zero occurs at x = −1. The graph looks almost linear at this point. This is a single zero of multiplicity 1. The last zero occurs at x = 4. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.
Intermediate Value Theorem Let f be a polynomial function. The Intermediate Value Theorem states that if f (a) and f (b) have opposite signs, then there exists at least one value c between a and b for which f (c) = 0.
EX. Show that the function f (x) = x 3 − 5x 2
Try It # Given the graph below, write a formula for the function shown.
