College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 2, Functions - Section 2.6 - Transformations of Functions - 2.6 Exercises: 47

Answer

Refer to the blue graph below.
1511924068

Work Step by Step

The given function can be written as: $f(x) = -\frac{1}{2}(x-1)^2+3$ RECALL: (1) The function $y=f(x-h)$ involves a horizontal shift of $h$ units to the right of the parent function $f(x)$ when $h\gt0$. The function involves a horizontal shift of $|h|$ units to the left when $h \lt0$. (2) The function $y=-f(x)$ involves a reflection about the x-axis of the parent function $f(x)$. (3) The function $y=a\cdot f(x)$ involves a vertical stretch of the parent function $f(x)$ when $a\gt1$, and involves a vertical compression when $0 \lt a \lt 1$. (4) The function $y=f(x)+c$ involves a vertical shift of $c$ units upward of the parent function $f(x)$ when $c \gt 0$. The function will involve a vertical shift of $|c|$ units downward when $c \lt 0$. The parent function of the given function is $y=x^2$. The given function can be written $y=-\frac{1}{2}f(x-1)+3$ where $f(x)$ is the parent function. Thus, the graph of the given function involves: (i) a horizontal shift of $1$ units to the right, (ii) a reflection about the x-axis, (iii) a vertical compression, and (iv) a shift of $3$ units upward of the parent function $f(x)=x^2$. Therefore, to graph the given function: (1) Graph the parent function $y=x^2$ . (Refer to the red graph in the attached image below) (2) Shift the graph of the parent function 1 unit to the right. (Refer to the orange graph in the attached image below.) (3) Reflect the graph in Step (2) 3 about the x-axis. (Refer to the green graph in the attached image below) (4) Multiply $\frac{1}{2}$ to each y-value of the graph in Step (3) while retaining the x-coordinate of each point. (Refer to the black graph in the attached image below.) (5) Shift the graph in Step (4) 3 units upward. (Refer to the blue graph in the attached image in the answer part above.)
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