Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 3 - Linear and Quadratic Functions - 3.3 Quadratic Functions and Their Properties - 3.3 Assess Your Understanding - Page 145: 20

Answer

$D$

Work Step by Step

RECALL: (1) The graph of the quadratic function $f(x)=ax^2=bx+c$ opens: (i) upward when $a \gt0$ and has its vertex as its minimum; or (ii) downward when $a\lt0$ and has its vertex as its maximum. (2) The vertex of the quadratic function $f(x)=ax^2=bx+c$ is at $\left(-\frac{b}{2a}, f(-\frac{b}{2a})\right)$ Let's compare $f(x)=x^2+2x+2$ to $f(x)=ax^2+bx+c$. We can see that $a=1$, $b=2$, $c=2$. $a\gt0$, hence the graph opens up, hence it's vertex is a minimum. The minimum value is at $x=-\frac{b}{2a}=-\frac{2}{2\cdot(1)}=-1.$ Hence the minimum value is $f(-1)=(-1)^2+2(-1)+2=1.$ Thus, the vertex is at $(-1,1)$ and the graph opens up. Therefore, the graph of the given function is the one in graph $D$.
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