Precalculus (10th Edition)

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

Chapter 2 - Functions and Their Graphs - 2.5 Graphing Techniques: Transformations - 2.5 Assess Your Understanding - Page 103: 15

Answer

$F$

Work Step by Step

RECALL: (1) The graph of $y=f(x-h)$ involves a horizontal shift of $|h|$ units (to the right when $h \gt 0$, to the left when $h\lt0$) of the parent function $f(x)$. (2) The graph of $y=f(x)+k$ involves a vertical shift of $|k|$ units (upward when $k \gt 0$, downward when $k\lt0$) of the parent function $f(x)$. (3) The graph of $y=a \cdot f(x-h)$ involves a vertical stretch or compression (stretch when$a\gt1$, compression when $0\lt a \lt1$) of the parent function $f(x)$. (4) The graph of $y=-f(x)$ involves a reflection about the $x$-axis of the parent function $f(x)$. The given graph is a parabola so its parent function is $f(x)=x^2$, the graph of which is a parabola that opens upward. (Refer to the attached image below for the graph of $f(x)=x^2$). The given graph shows that the graph of the parent function was reflected about the $x$-axis and shifted $2$ units to the left. Use the rules listed above to find the equation of the given graph. (1) Reflecting $f(x)=x^2$ about the $x$-axis (Rule (4) above) makes the equation of the resulting function $y=-f(x) = -x^2$. (2) Shifting the graph horizontally $2$ units to the left (Rule (2) above) makes the equation of the resulting function $y=-(x+2)$. Therefore the answer is $F$.
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