Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Section 3.1 - Quadratic Functions and Models - 3.1 Exercises - Page 252: 27


a. $f(x)=3(x-1)^{2}-2$ b. see image below c. Minimum value $f(1)=-2$

Work Step by Step

a. Complete the square: $3x^{2}-6x=3(x^{2}-2x)= 3(x^{2}-2\cdot(x)(1)+1^{2}-1^{2})$ $=(x-1)^{2}-3$ $f(x)=3(x-1)^{2}-3+1$ $f(x)=3(x-1)^{2}-2$ b. To sketch, begin with the parent function $f_{1}(x)=x^{2},$ (blue, dashed line in the image) and, since $f(x)=f_{1}(x-1)-2,$ shift the graph to the right by $1$ units, and down $2$ units (solid red line, see image) c. The graph of $f(x)=a(x-h)^{2}+k$ is a parabola, and, if $a > 0$, then the quadratic function $f$ opens upward and has the minimum value $k$ at $x=h=-\displaystyle \frac{b}{2a}$. Minimum value$:\quad f(1)=-2$
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