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: 18


please see "step by step"

Work Step by Step

Rewrite f(x) in standard form, $f(x)=a(x-h)^{2}+k$, read the vertex, (h,x) For the y-intercept, calculate f(0) For the x- intercept, solve f(x) = 0 for x. If $a>0$, parabola opens up, the vertex is a minimum point, If $a<0$, parabola opens down, the vertex is a maximum. With this information (and possible additional points) sketch a graph Read the graph for range and domain. ------------------ a. factor out $-1$, $ f(x)=-(x^{2}+4x) +4 \quad$... complete the square $f(x)=-(x^{2}+3(3)x+2^{2}-2^{2}) +4$ $f(x)=-(x+2)^{2}+4+4$ $f(x)=-(x+2)^{2}+8$ b. vertex: $(h,k)=(-2, 8)$, a=$-1$, opens down, the vertex is a maximum y-intercept: f(0) = $4$ x-intercepts: f(x)=0 $-(x+2)^{2}+8=0$ $(x+2)^{2}=8\qquad/\sqrt{..}$ $x+2=\pm 2\sqrt{2}$ $x=-2\pm 2\sqrt{2}$ c. see image (one pair of additional points, either side of the vertex). d. domain: all reals, $\mathbb{R}$ range: $(-\infty,8]$
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