Stitz-Zeager_College_Algebra_e-book

If the towers are 100 feet tall nd the height of the

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Unformatted text preview: t; 0. 4 Up if k > 0, down if k < 0 5 Actually, we could also take the standard form, f (x) = a(x − h)2 + k, expand it, and compare the coefficients of it and the general form to deduce the result. However, we will have another use for the completed square form of the general form of a quadratic, so we’ll proceed with the conversion. 3 2.3 Quadratic Functions 143 we have derived the vertex formula for the general form as well. Note that the value a plays the exact same role in both the standard and general equations of a quadratic function − it is the coefficient of x2 in each. No matter what the form, if a > 0, the parabola opens upwards; if a < 0, the parabola opens downwards. Now that we have the completed square form of the general form of a quadratic function, it is time to remind ourselves of the quadratic formula. In a function context, it gives us a means to find the zeros of a quadratic function in general form. Equation 2.5. The Quadratic Formula: If a, b, c are real numbers with a = 0, then the solutions to ax2 + bx + c = 0 are √ −b ± b2 − 4ac x= . 2a Assuming the conditions of Equation 2.5, the solutions to ax2 + bx + c = 0 are precisely the zeros of f (x) = ax2 + bx + c. We have shown an equivalent formula for f is f (x) = a x + b 2a 2 + 4ac − b2 . 4a Hence, an equation equivalent to ax2 + bx + c = 0 is a x+ b 2a 2 + 4ac − b2 = 0. 4a Solving gives a x+ b 2a 2 + 4ac − b2 4a a x+ 2 b 2a 1 b a x+ a 2a b x+ 2a x+ =0 =− 2 4ac − b2...
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