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# Class Note for MATH 1314 with Professor Heeth at UH

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
Math 1314 Lesson 12 Curve Sketching One of our objectives in this part of the course is to be able to graph functions In this lesson we ll add to some tools we already have to be able to sketch an accurate graph of each function From prerequisite material we can nd the domain y intercept and end behavior of the graph of a function and from the last two sections we can learn much about a function by analyzing the first and second derivatives We also know how to find the zeros of some functions We ll expand that group of function before we continue to curve sketching The Rational Zeros of a Polynomial Function The rational zeros of a function are the zeros of the function that can be written as a fraction such as 2 or f Sometimes we can find the rational zeros of a function by factoring Example 1 Find the rational zeros fx x3 7 16x Example 2 Find the rational zeros fx x4 7 11x 18 Note that zeros that are square roots are NOT rational roots Imaginary solutions to the equation fx 0 such as i3z39 are NOT rational roots Sometimes we won t be able to factor the function Then we ll need another method We ll use a theorem called the Rational Zeros Theorem First we ll nd all of the possible rational zeros of a given function Then we can use a calculator or synthetic diVision to determine which 7 if any 7 of the possible rational zeros are actually zeros of the function Here s the theorem Rational Zeros Theorem Suppose x anxquot an1xquot391 alx do where an 72 0 and a0 72 0 and all ofthe coefficients of the polynomial are integers If x E is a rational zero of the function where p and q have no common factors then p is a q factor of the constant term a0 and q is a factor of the leading coefficient an Example 3 Find the possible rational zeros of fx 2x3 7 3x2 8x 7 4 Example 4 Find all rational zeros of fx x3 6x2 3x 7 10 or state that there are none Example 5 Find all rational zeros of fx x3 7x2 16x 12 or state that there are none Example 6 Find all rational zeros of fx x4 7 4x3 3x2 4x 7 4 or state that there are none Example 7 Find all rational zeros of fx x3 7 3x2 7 l or state that there are none Example 8 Find all rational zeros of fx x4 7 5x3 4x2 or state that there are none Example 9 Find all rational zeros of fx x3 4x2 4x 16 or state that there are none Curve Sketching Now we ll turn our attention to graphing functions You will need to be able to use the following guide to sketch the graphs of functions A Guide to Curve Sketching 1 Determine the domain off 2 Find the rational xintercepts and y intercept of the function If there are no rational x intercepts say so 3 Determine the end behavior of the function 4 For an exponential function determine any horizontal asymptotes 5 Determine where the function is increasing and where it is decreasing 6 Find the x and y coordinates of any relative extrema 7 Determine where the function is concave upward and where it is concave downward 8 Find the x and y coordinates of any points of in ection 9 If necessary plot a few additional points to determine the shape of the graph 10 Sketch the function Recall the generalizations about end behavior of a polynomial function from College Algebra PEHH NELL POLH NOHL Example 10 Use the guide to curve sketching to sketch f x x4 4x3 Sometimes a function has some zeros that are not rational We may occasionally give you the approximate zeros of the function and ask you to complete the rest of the guide to curve sketching Example 11 Use the guide to curve sketching to sketch x x3 6x2 7 15x 3 Note The approximate zeros of the function are 022 172 and 7 94 Example 12 Use the guide to curve sketching to sketch x x3 7x2 16x 12 Note we found the rational zeros in example 5 Example 13 Use the guide to curve sketching to sketch x x3 7 8x2 19x 7 12 Example 14 Use the guide to curve sketching to sketch x xex For the next two problems you are given all of the information listed in the guide to curve sketching You just need to use it to graph the function Example 15 Sketch the function if you are given the following information xintercept 0 0 4 0 yintercept 0 0 end behavior T T relative minimum 3 27 increasing intervals 3 00 decreasing intervals 00 0 and 0 3 points of in ections 0 0 and 2 l6 concave upward intervals 00 0 and 2 0 concave downward intervals 0 2 Example 16 Sketch the function if you are given the following information xintercept 0 0 l 0 and 2 0 yintercept 0 0 end behavior T T relative maximum 061 020 relative minimum 164 062 and 0 0 increasing intervals 164 061 and 0 co decreasing intervals oo 164 and 061 0 points of in ections 027 009 and 123 027 concave upward intervals oo 123 and 027 co concave downward intervals 123 027 Example 17 Here is the graph of a polynomial function True or false The function has three zeros The gmph of the function is increasing on one interval and decreasing on tWo intervals The gmph of the function has one relative maximum and one relative The gmph of the function has tWo in ection points The function could be a quartic function 43911 degree With a positive leading coef cient M P Nt From this section you should be able to Find any rational zeros of a 3rd or 4 11 degree polynomial Use the guide to curve sketching to sketch the graph of apolynomial or exponential Sketch a graph of a function given all of the information from the guide to curve sketching Answer questions about the gmph of a function given the gmph of the function

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