3 Graph f(x) ax2 bx c by Completing the Square Procedure Rewriting f(x) ax2 bx c in the Form f(x) a(x h)2 k by Completing the Square Step 1: The coeffi cient of the square term must be 1. If it is not 1, multiply or divide both sides of the equation (including f (x)) by the appropriate value to obtain a leading coeffi cient of 1. Step 2: Separate the constant from the terms containing the variables by grouping the variable terms with parentheses. Step 3: Complete the square for the quantity in the parentheses. Find half of the linear coeffi cient, then square the result. Add that quantity inside the parentheses, and subtract the quantity from the constant. (Adding and subtracting the same number on the same side of an equation is like adding 0 to the equation.) Step 4: Factor the expression inside the parentheses. Step 5: Solve for f(x). EXAMPLE 6 Graph each function. Begin by completing the square to rewrite each function in the form f (x) a(x h)2 k. Include the intercepts. a) f (x) x2 6x 10 b) g(x) 1 2 x2 4x 6 Solution a) Step 1: The coeffi cient of x2 is 1. Step 2: Separate the constant from the variable terms using parentheses. f (x) (x2 6x) 10 Step 3: Complete the square for the quantity in the parentheses. 1 2 (6) 3 32 9 Add 9 inside the parentheses, and subtract 9 from the 10. This is like adding 0 to the equation. f (x) (x2 6x 9) 10 9 f (x) (x2 6x 9) 1 Step 4: Factor the expression inside the parentheses. f (x) (x 3)2 1 Step 5: The equation is solved for f (x). www.mhhe.com/messersmith SECTION 10.5 Quadratic Functions and Their Graphs 659
messersmith_power_intermediate_algebra_1e_ch4_7_10
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