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: 28


a) In standard form, $f(x) = 5(x+3)^2-41$. b) Graph is attached. c) $(-3, -41)$ is the minimum point.

Work Step by Step

a) To find the standard form of some function $f(x) = ax^2+bx+c$, the standard form would be $f(x) = a(x-h)^2+k$ where $h = -\frac{b}{2a}$ and $k = f(h)$. This is a standard result derived in the book simplifies the algebra and gives a closed form for the end result. For this problem, $a= 5,b = 30, c = 4$. Plugging above, we get $h = -3, k = f(-3) = -41$ and hence $f(x) = 5(x+3)^2-41.$ b) To plot the graph, we plot the vertex and the y-intercept and join them using a smooth curve; we also note that there would be a line of symmetry about a vertical axis passing through this vertex. This allows us to be able to draw the parabola using only two points. The vertex can be easily deduced from the standard form; it is the point $(h, k)$ so in this case $(-3, -41)$. The y-intercept can be easily deduced from the original form; it is the $c$ (this can be verified by plugging in $x=0$ into $f(x)$ and seeing that it reduces to $c$). Therefore, in this case, the y-intercept is $(0, 4)$. c) 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 $(-3, -41)$ is a minimum.
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