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Unformatted text preview: Section 5.2 Vertex Form 441 Version: Fall 2007 5.2 Vertex Form In the previous section, you learned that it is a simple task to sketch the graph of a quadratic function if it is presented in vertex form f ( x ) = a ( x − h ) 2 + k. (1) The goal of the current section is to start with the most general form of the quadratic function, namely f ( x ) = ax 2 + bx + c, (2) and manipulate the equation into vertex form . Once you have your quadratic function in vertex form, the technique of the previous section should allow you to construct the graph of the quadratic function. However, before we turn our attention to the task of converting the general quadratic into vertex form, we need to review the necessary algebraic fundamentals. Let’s begin with a review of an important algebraic shortcut called squaring a binomial . Squaring a Binomial A monomial is a single algebraic term, usually constructed as a product of a number (called a coefficient ) and one or more variables raised to nonnegative integral powers, such as − 3 x 2 or 14 y 3 z 5 . The key phrase here is “single term.” A binomial is an algebraic sum or difference of two monomials (or terms), such as x + 2 y or 3 ab 2 − 2 c 3 . The key phrase here is “two terms.” To “square a binomial,” start with an arbitrary binomial, such as a + b , then multiply it by itself to produce its square ( a + b )( a + b ), or, more compactly, ( a + b ) 2 . We can use the distributive property to expand the square of the binomial a + b . ( a + b ) 2 = ( a + b )( a + b ) = a ( a + b ) + b ( a + b ) = a 2 + ab + ba + b 2 Because ab = ba , we can add the two middle terms to arrive at the following property. Property 3. The square of the binomial a + b is expanded as follows. ( a + b ) 2 = a 2 + 2 ab + b 2 (4) Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 442 Chapter 5 Quadratic Functions Version: Fall 2007 ⚏ Example 5. Expand ( x + 4) 2 . We could proceed as follows. ( x + 4) 2 = ( x + 4)( x + 4) = x ( x + 4) + 4( x + 4) = x 2 + 4 x + 4 x + 16 = x 2 + 8 x + 16 Although correct, this technique will not help us with our upcoming task. What we need to do is follow the algorithm suggested by Property 3 . Algorithm for Squaring a Binomial. To square the binomial a + b , proceed as follows: 1. Square the first term to get a 2 . 2. Multiply the first and second terms together, and then multiply the result by two to get 2 ab . 3. Square the second term to get b 2 . Thus, to expand ( x + 4) 2 , we should proceed as follows. 1. Square the first term to get x 2 2. Multiply the first and second terms together and then multiply by two to get 8 x . 3. Square the second term to get 16. Proceeding in this manner allows us to perform the expansion mentally and simply write down the solution....
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