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: 19


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. $f(x)=(2x^{2}+4x) +3\quad $ factor out $2$, $ f(x)=2(x^{2}+2x) +3 \quad$... complete the square $f(x)=2(x^{2}+2(1)x+1^{2}-1^{2})+3$ $f(x)=2(x+1)^{2}-2+3$ $f(x)=2(x+1)^{2}+1$ b. vertex: $(h,k)=(-1, 1)$, a=$+2$, opens up, the vertex is a minimum y-intercept: f(0) = $3$ x-intercepts: f(x)=0 $(x-1)^{2}+1=0$ $(x-1)^{2}=-1$ (square can not be negative, no solutions) x-intercepts: none c. see image (two pairs of additional points, either side of the vertex). d. domain: all reals, $\mathbb{R}$ range: $[1,\infty)$
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