Algebra 2 (1st Edition)

Published by McDougal Littell
ISBN 10: 0618595414
ISBN 13: 978-0-61859-541-9

Chapter 4 Quadratic Functions and Factoring - 4.7 Complete the Square - 4.7 Exercises - Quiz for Lessons 4.5-4.7 - Page 291: 18


The vertex form of the function is $y=(x-(-\displaystyle \frac{9}{2}))^{2}+(-\frac{5}{4}).$ The vertex is $(-\displaystyle \frac{9}{2},-\frac{5}{4})$.

Work Step by Step

$ y=x^{2}+9x+19\qquad$ ...write in form of $x^{2}+bx=c$ (add $-19$ to each side). $ y-19=x^{2}+9x\qquad$ ...square half the coefficient of $x$. $(\displaystyle \frac{9}{2})^{2}=\frac{81}{4}\qquad$ ...complete the square by adding $\displaystyle \frac{81}{4}$ to each side of the expression $ y-19+\displaystyle \frac{81}{4}=x^{2}+9x+\frac{81}{4}\qquad$ ... write $x^{2}+9x+\displaystyle \frac{81}{4}$ as a binomial squared. $ y+\displaystyle \frac{5}{4}=(x+\frac{9}{2})^{2}\qquad$ ...add $-\displaystyle \frac{5}{4}$ to each side of the expression $ y=(x+\displaystyle \frac{9}{2})^{2}-\frac{5}{4}\qquad$ ...write in vertex form $y=a(x-h)^{2}+k$. $y=(x-(-\displaystyle \frac{9}{2}))^{2}+(-\frac{5}{4})$ The vertex form of a quadratic function is $y=a(x-h)^{2}+k$ where $(h,k)$ is the vertex of the function's graph. Here, $h=-\displaystyle \frac{9}{2},\ k=-\displaystyle \frac{5}{4}$, so the vertex is $(-\displaystyle \frac{9}{2},-\frac{5}{4})$
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