Chapter 3 Section 1

\[ \definecolor{ucrblue}{rgb}{0.0627,0.3843,0.6039} \definecolor{ucrgold}{rgb}{0.9686,0.6941,0.2824} \definecolor{ucrred}{rgb}{0.8941,0,0.1686} \definecolor{ucrgreen}{rgb}{0.4706,0.7451,0.1255} \definecolor{ucraccent}{rgb}{1.0000,0.9569,0.8392} \DeclareMathOperator*{\LO}{O} \DeclareMathOperator*{\Lo}{o} \DeclareMathOperator*{\Recip}{Recip} \DeclareMathOperator*{\abs}{abs} \DeclareMathOperator{\pow}{pow} \]

Notes

• Notes

Linguistic Mapping Exercises

• Linguistic Mapping Exercises PDF

Knowledge Checks

1. Take \(f\) to be the function that is given by \[f(x) = 6(x-2)^2 - 15.\]
   1. Determine the axis of symmetry and vertex of \(f\).
   2. Find all roots (\(x\)-intercepts) of \(f\).
   3. Sketch \(f\).
   4. Solve \(f>0\).
2. Find and classify the extremal points of \(f\), where \[f(x) = -10 x^2 + 10 x + \tfrac{1}{2}.\]
3. Find polynomials \(q\) and \(r\) so that \(\frac{r(x)}{x - 4}\) is a proper fraction and \[5x^3+10x^2-5 = q(x)(x - 4) + r(x).\]
4. Write \(4 x^3 - 32 x^2 - 212 x + 240\) as a product of linear factors given that \(x=-5\) is a root or zero.
5. Take \(f\) to be the polynomial function that is given by \[f(x) = 20(x+11)^5(x+6)^3(x-1)^2(x-3)^4(2x-7)^5.\] List the zeros of \(f\) together with their orders.
6. Determine the leading term of the polynomial \(f\) that is given by \[f(x) = 20(x+11)^5(x+6)^3(x-1)^2(x-3)^4(2x-7)^5.\] Describe the asymptotic behavior of \(f\).
7. Sketch the polynomial \(f\), where \[f(x) = 20(x+11)^5(x+6)^3(x-1)^2(x-3)^4(2x-7)^5.\]

• Knowledge Checks PDF
• Knowledge Checks Solutions

Remember, do the knowledge checks before checking the solutions.

Practice Problems

• Practice Problems

Return

• Return

© Copyright 2025 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.