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# Class Note for MATH 1314 with Professor Gross at UH 2

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Date Created: 02/06/15
M1314 Lesson 12 1 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 1 as 2 or Somet1mes we can find the ratlonal zeros of a functlon by factor1ng Example 1 Find the rational zeros fx x3 16x Example 2 Find the rational zeros fx x4 11x2 18 M1314 Lesson 12 2 Note that zeros that are square roots are NOT rational roots Imaginary solutions to the equation fx 0 such as i 3139 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 using the Rational Zeros Theorem 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 fx anxquot a x 1 a0 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 3x2 8x 4 Example 4 Find all rational zeros of f x x3 6x2 3x 10 or state that there are none M1314 Lesson 12 3 Example 5 Find all rational zeros of f x x3 7x2 l6x 12 or state that there are none Example 6 Find all rational zeros of f x x4 4x3 3x2 4x 4or state that there are none M1314 Lesson 12 Example 7 Find all rational zeros of f x x3 3x2 l or state that there are none Example 8 Find all rational zeros of f x x4 5x3 4x2 or state that there are none M1314 Lesson 12 5 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 x intercepts and y intercept of the inction If there are no rational x intercepts say so 3 Determine the end behavior of the function 4 For an exponential inction determine any horizontal asymptotes 5 Determine where the inction is increasing and where it is decreasing 6 Find the x andy coordinates of any relative extrema 7 Determine where the inction is concave upward and where it is concave downward 8 Find the x andy coordinates of any points of in ection 9 If necessary plot a few additional points to determine the shape ofthe graph 10 Sketch the function Recall the generalizations about end behavior of a polynomial inction from College Algebra Evendegree polynomials look like Odddegree polynomials look like yix2 yix3 M1314 Lesson 12 Example 10 Use the guide to curve sketching to sketch f x x4 4x3 M1314 Lesson 12 7 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 fx x3 6x2 15x 3 Note the approximate zeros of the function are 022 172 and 7 94 A V M1314 Lesson 12 8 Example 12 Use the guide to curve sketching to sketch fx x3 7x2 16x 12 Note we found the rational zeros in example 5 M1314 Lesson 12 Example 13 Use the guide to curve sketching to sketch fx x3 8x2 19x 12 M1314 Lesson 12 Example 14 Use the guide to curve sketching to sketch f x xe 10 M1314 Lesson 12 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 X intercept 0 0 4 0 y intercept 0 0 end behavior T T relative minimum 3 27 increasing intervals 3 co decreasing intervals 000 03 points of in ections 0 0 2 l6 concave upward intervals 00 0 and 2 co concave downward intervals 0 2 M1314 Lesson 12 Example 16 Sketch the function if you are given the following information X intercepts 0 0 1 0 and 2 0 y intercept 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 12 M1314 Lesson 12 13 Example 17 Here is the graph of a polynomial function Which of the statements below isare true The function has three zeros The graph of the function is increasing on one interval and decreasing on two intervals The graph of the function has one relative maximum and one relative minimum The graph of the function has two in ection points The function could be a quartic function 4 11 degree with a positive leading coefficient V eP Nf From this section you should be able to Find any rational zeros of a 3ml or 4th degree polynomial Use the guide to curve sketching to sketch the graph of a polynomial or exponential Sketch a graph of a function given all of the information from the guide to curve sketching Answer questions about the graph of a function given the graph of the function

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