Lesson 12-filled - Math 1314 Lesson 12 Curve Sketching One...

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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 find 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 . 5 1 - Sometimes we can find the rational zeros of a function by factoring. Example 1: Find the rational zeros: . 16 ) ( 3 x x x f - = Example 2 : Find the rational zeros: . 18 11 ) ( 2 4 + - = x x x f
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Note that zeros that are square roots are NOT rational roots. Imaginary solutions to the equation 0 ) ( = x f (such as i 3 ± ) 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 find 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 – if any – of the possible rational zeros are actually zeros of the function.
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