College Algebra (10th Edition)

Published by Pearson

Chapter 3 - Section 3.5 - Graphing Techniques: Transformations - 3.5 Assess Your Underdstanding: 29

Answer

$y=-\sqrt{x+3}+2$

Work Step by Step

RECALL: (i) $y=-f(x)$ involves a reflection about the x-axis of the parent function $y=f(x)$. (ii) $y=f(-x)$ involves a reflection about the y-axis of the parent function $y=f(x)$. (iii) $y=af(x)$ involves either a vertical compression by a factor of $a$ of the parent function $f(x)$ when $a\gt 1$ or a vertical stretch when $0 \lt a \lt 1$. (iv) $y=f(x-h)$ involves a horizontal shift of either $h$ units to the right of the parent function $f(x)$ when $h \gt 0$ or $|h|$ units to the left when $h \lt0$. (v) $y=f(x) + k$ involves either a vertical shift of $k$ units upward of the parent function when $k\gt 0$ or $|k|$ units downward when $k \lt0$. The graph involves the following transformations of the parent function $=\sqrt{x}$: (1) Reflection about the x-axis: Using (1) above gives: $y=-f(x) \\y=-\sqrt{x}$ (2) Shifted up 2 units. Using (v) above gives: $y=f(x)+k$ where $k=2$. Thus, the equation is: $\\y=-\sqrt{x}+2$ (3) Shifted to the left by $3$ units: Using (iv) above gives: $y=f(x-h)$ where $h=-3$. Thus, the equation is $\\y=-\sqrt{x-(-3)}+2 \\y=-\sqrt{x+3}+2$ Therefore, the function is: $y=-\sqrt{x+3}+2$

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