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Check all symmetries that apply.
(a)
28x^(2)+32y^(2)=65
(b)
y=-7x
Symmetry:

x-axis

y-axis
origin
none of the above
Symmetry:

x-axis

y-axis
origin
none of the above

Check all symmetries that apply. $\newline$ (a) $28 x^{2}+32 y^{2}=65$ $\newline$ (b) $y=-7 x$ $\newline$ Symmetry: $\newline$ $x$ -axis $\newline$ $y$ -axis $\newline$ origin $\newline$ none of the above $\newline$ Symmetry: $\newline$ $x$ -axis $\newline$ $y$ -axis $\newline$ origin $\newline$ none of the above

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Reflections of functions

Full solution

Q. Check all symmetries that apply.

\newline

(a)

28 x^{2}+32 y^{2}=65

\newline

(b)

y=-7 x

\newline

Symmetry:

\newline

x

-axis

\newline

y

-axis

\newline

origin

\newline

none of the above

\newline

Symmetry:

\newline

x

-axis

\newline

y

-axis

\newline

origin

\newline

none of the above

Replace $y$ with $-y$ : To determine if the equation (a) $28x^2 + 32y^2 = 65$ has x-axis symmetry, replace $y$ with $-y$ and see if the equation remains unchanged. $\newline$ Calculation: $28x^2 + 32(-y)^2 = 28x^2 + 32y^2 = 65$
X-axis symmetry confirmed: Since the equation is unchanged when $y$ is replaced with $-y$ , the equation $(a)$ has $x$ -axis symmetry.
Replace $x$ with $-x$ : To determine if the equation (a) has y-axis symmetry, replace $x$ with $-x$ and see if the equation remains unchanged. $\newline$ Calculation: $28(-x)^2 + 32y^2 = 28x^2 + 32y^2 = 65$
Y-axis symmetry confirmed: Since the equation is unchanged when $x$ is replaced with $-x$ , the equation (a) has $y$ -axis symmetry.
Replace $x$ and $y$ with negatives: To determine if the equation (a) has origin symmetry, replace $x$ with $-x$ and $y$ with $-y$ and see if the equation remains unchanged. $\newline$ Calculation: $28(-x)^2 + 32(-y)^2 = 28x^2 + 32y^2 = 65$
Origin symmetry confirmed: Since the equation is unchanged when both $x$ and $y$ are replaced with their negatives, the equation $(a)$ has origin symmetry.
Replace $y$ with $-y$ : To determine if the equation (b) $y = -7x$ has x-axis symmetry, replace $y$ with $-y$ and see if the equation remains unchanged. $\newline$ Calculation: $-y = -7x$
No x-axis symmetry: Since the equation is not unchanged (it becomes $-y = -7x$ , which is not equivalent to $y = -7x$ ), the equation $(b)$ does not have x-axis symmetry.
Replace $x$ with $-x$ : To determine if the equation (b) has y-axis symmetry, replace $x$ with $-x$ and see if the equation remains unchanged. $\newline$ Calculation: $y = -7(-x) = 7x$
No y-axis symmetry: Since the equation is not unchanged (it becomes $y = 7x$ , which is not equivalent to $y = -7x$ ), the equation $(b)$ does not have y-axis symmetry.
Replace $x$ and $y$ with negatives: To determine if the equation (b) has origin symmetry, replace $x$ with $-x$ and $y$ with $-y$ and see if the equation remains unchanged. $\newline$ Calculation: $-y = -7(-x) = 7x$
Origin symmetry confirmed: Since the equation is unchanged when both $x$ and $y$ are replaced with their negatives (it becomes $-y = 7x$ , which is equivalent to $y = -7x$ after multiplying both sides by $-1$ ), the equation (b) has origin symmetry.

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4

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\newline

7

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\newline

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