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$Rectangle WXYZ with vertices W(-7,2),X(5,2),Y(5,-6), and Z(-7,-6) : a) dilation with scale factor of 1//4 using (-7,6) as the center b) reflection in the y-axis {:[W^(')(◻)],[X^(')(◻)],[Y^(')(◻)],[Z^(')(◻)]:}$

$6$ . Rectangle $W X Y Z$ with vertices $W(-7,2), X(5,2), Y(5,-6)$ , and $Z(-7,-6)$ : $\newline$ a) dilation with scale factor of $1 / 4$ using $(-7,6)$ as the center $\newline$ b) reflection in the $y$ -axis $\newline$ $\begin{array}{l} W^{\prime}(\square) \\ X^{\prime}(\square) \\ Y^{\prime}(\square) \\ Z^{\prime}(\square) \end{array}$

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Math Problems

Algebra 2

Write equations of cosine functions using properties

Full solution

6

. Rectangle

W X Y Z

with vertices

W(-7,2), X(5,2), Y(5,-6)

, and

Z(-7,-6)

\newline

a) dilation with scale factor of

1 / 4

using

(-7,6)

as the center

\newline

b) reflection in the

y

-axis

\newline

\begin{array}{l} W^{\prime}(\square) \\ X^{\prime}(\square) \\ Y^{\prime}(\square) \\ Z^{\prime}(\square) \end{array}

Dilation Center Subtraction: For dilation, subtract the center of dilation $(-7,6)$ from each vertex coordinate. $\newline$ $W' = W - \text{center} = (-7,2) - (-7,6) = (0,-4)$ $\newline$ $X' = X - \text{center} = (5,2) - (-7,6) = (12,-4)$ $\newline$ $Y' = Y - \text{center} = (5,-6) - (-7,6) = (12,-12)$ $\newline$ $Z' = Z - \text{center} = (-7,-6) - (-7,6) = (0,-12)$
Scale Factor Multiplication: Multiply each coordinate by the scale factor $\frac{1}{4}$ . $\newline$ $W'' = (0 \times \frac{1}{4}, -4 \times \frac{1}{4}) = (0, -1)$ $\newline$ $X'' = (12 \times \frac{1}{4}, -4 \times \frac{1}{4}) = (3, -1)$ $\newline$ $Y'' = (12 \times \frac{1}{4}, -12 \times \frac{1}{4}) = (3, -3)$ $\newline$ $Z'' = (0 \times \frac{1}{4}, -12 \times \frac{1}{4}) = (0, -3)$
Center Addition: Add the center of dilation back to each vertex coordinate. $\newline$ $W^{\prime\prime\prime}$ = $W^{\prime\prime}$ + center = $(0, -1)$ + $(-7, 6)$ = $(-7, 5)$ $\newline$ $X^{\prime\prime\prime}$ = $X^{\prime\prime}$ + center = $(3, -1)$ + $(-7, 6)$ = $(-4, 5)$ $\newline$ $W^{\prime\prime}$ $0$ = $W^{\prime\prime}$ $1$ + center = $W^{\prime\prime}$ $2$ + $(-7, 6)$ = $W^{\prime\prime}$ $4$ $\newline$ $W^{\prime\prime}$ $5$ = $W^{\prime\prime}$ $6$ + center = $W^{\prime\prime}$ $7$ + $(-7, 6)$ = $W^{\prime\prime}$ $9$
Y-Axis Reflection: For reflection in the y-axis, change the sign of the $x$ -coordinates. $\newline$ $W' = (-7, 5)$ reflected is $(7, 5)$ $\newline$ $X' = (-4, 5)$ reflected is $(4, 5)$ $\newline$ $Y' = (-4, 3)$ reflected is $(4, 3)$ $\newline$ $Z' = (-7, 3)$ reflected is $(7, 3)$