Question

NO LINKS!! Please help me with this symmetry part 2​

Answer 1

`x=(1)/(8)y^2`

`x=-y^2+9`

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.

Therefore, any function that includes the term y² will be symmetric with respect to the x-axis since (-y)² = y².

`\begin{aligned}&\textsf{Given}: \quad& x&=(1)/(8)y^2\\&\textsf{Replace $y$ for $(-y)$}: \quad& x&=(1)/(8)(-y)^2\\&\textsf{Simplify}: \quad &x&=(1)/(8)y^2\\\end{aligned}`

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

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

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

Answer 2

Functions that are symmetric with respect to the x-axis are those with even degree of y, i.e y², y⁴ etc.

Correct choices are:

• x = 1/8y²
• x = - y² + 9