College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 3 - Section 3.5 - Graphing Techniques: Transformations - 3.5 Assess Your Underdstanding: 51


Refer to the blue graph below.

Work Step by Step

Graph the parent function $ y=x^2$. (refer to green graph in the the attached image below) RECALL: (1) The function $y=f(x-h)$ involves a horizontal shift of either $h$ units to the right of the parent function $f(x)$ if $h \gt 0$, or $|h|$ units to the left when $h \lt0$ (2) The function $y=a \cdot f(x)$ involves either a vertical stretch by a factor of $a$ of the parent function $f(x)$ when $a \gt 1$, or a vertical compressions when $0 \lt a \lt 1$. (3) The function $y=f(x)+k$ involves a vertical shift of either $k$ units upward of the parent function when $k\gt0$, or $|k|$ units downward when $k\lt0$. The given function involves all transformations mentioned in (1), (2), and (3) above with $h=-1$, $a=2$, and $k=-3$. Note that following the order of operation, the first transformation to be applied is the subtraction of $h$ from $x$, followed by the multiplication of $2$ to $f(x)$, and the subtraction of $3$ is last. Thus, to graph the given function, perform the following: (i) Shift each plotted point of the parent function $y=x^2$ one unit to the left (refer to the black graph in the attached image below); and (ii) For each plotted of the resulting graph in (i), double the y-value while retaining the $x$ value. (refer to the red graph in the image below). (iii) Shift each point of the plotted points in (ii) three units downward. (Refer to the attached image below in the answer part above.)
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