College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.1 - Quadratic Functions and Models - 3.1 Exercises - Page 288: 44

Answer

$(2, -1)$ is the minimum point.

Work Step by Step

First, we find the standard form but before, let's expand and simplify. $g(x) = 2x(x-4)+7 = 2x^2-8x+7$ To find the standard form of some function $g(x) = ax^2+bx+c$, it would be $g(x) = a(x-h)^2+k$ where $h = -\frac{b}{2a}$ and $k = g(h)$. This is a standard result derived in the book simplifies the algebra and gives a closed form for the end result. For this problem, $a= 2,b = -8, c = 7$. Plugging above, we get $h = 2, k = g(2) = -1$ and hence $g(x) = 2(x-2)^2-1.$ The vertex can be easily deduced from the standard form; it is the point $(h, k)$ so in this case $(2, -1)$. The maximum or minimum of a quadratic function is attained at the vertex. To determine whether the vertex is a maximum or minimum, we look at $a$: - if $a>0$, this tells us the function will grow towards positive infinity and hence, the vertex is a minimum. - if $a<0$, this tells us that the function will grow towards negative infinity and hence, the vertex is a maximum. Looking at $a$ in this case tells us that the vertex $(2, -1)$ is a minimum.
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