Question

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Use the algebraic tests to check for symmetry with respect to both axes and the origin. (Select all that apply.)
x^2 - y = 9
1. x-axis symmetry
2. y-axis symmetry
3. origin symmetry
4. no symmetry

Answer 1

• 2. y-axis symmetry

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Given function:

• x² - y = 9

Rewrite it as:

• y = x² - 9

This is a quadratic function, translation of the parent function y = x² down by 9 units.

We know the graph of the quadratic function is parabola, with the y-axis symmetry.

See attached to confirm.

Answer 2

2. y-axis symmetry

Explanation:

Functions are symmetric with respect to the x-axis if for every point (a, b) on the graph, there is also a point (a, −b) on the graph:

• f(x, y) = f(x, −y)

To determine if a graph is symmetric with respect to the x-axis, replace all the y's with (−y). If the resultant expression is equivalent to the original expression, the graph is symmetric with respect to the x-axis.

`\begin{aligned}&\textsf{Given}: \quad &x^2-y&=9\\&\textsf{Replace $y$ for $(-y)$}: \quad &x^2-(-y)&=9\\&\textsf{Simplify}: \quad &x^2+y&=9\end{aligned}`

Therefore, since the resultant expression is not equivalent to the original expression, it is not symmetric with respect to the x-axis.

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Functions are symmetric with respect to the y-axis if for every point (a, b) on the graph, there is also a point (-a, b) on the graph:

• f(x, y) = f(-x, y)

To determine if a graph is symmetric with respect to the x-axis, replace all the x's with (−x). If the resultant expression is equivalent to the original expression, the graph is symmetric with respect to the y-axis.

`\begin{aligned}&\textsf{Given}: \quad &x^2-y&=9\\&\textsf{Replace $x$ for $(-x)$}: \quad &(-x)^2-y&=9\\&\textsf{Simplify}: \quad &x^2+y&=9\end{aligned}`

Therefore, since the resultant expression is equivalent to the original expression, it is symmetric with respect to the y-axis.

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Functions are symmetric with respect to the origin if for every point (a, b) on the graph, there is also a point (-a, -b) on the graph:

• f(x, y) = f(-x, -y)

To determine if a graph is symmetric with respect to the origin, replace all the x's with (−x) and all the y's with (-y). If the resultant expression is equivalent to the original expression, the graph is symmetric with respect to the origin.

`\begin{aligned}&\textsf{Given}: \quad &x^2-y&=9\\&\textsf{Replace $x$ for $(-x)$ and $y$ for $(-y)$}: \quad &(-x)^2-(-y)&=9\\&\textsf{Simplify}: \quad &x^2+y&=9\end{aligned}`

Therefore, since the resultant expression is not equivalent to the original expression, it is not symmetric with respect to the origin.