Question

NO LINKS!! Please help me with this symmetry. Select all that apply. ​

Answer 1

`y=-7x^2`

`y=8x^2-3`

Explanation:

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 y-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.

Therefore, any function that includes the term x² will be symmetric with respect to the x-axis since (-x)² = x².

`\begin{aligned}&\textsf{Given}: \quad& y&=-7x^2\\&\textsf{Replace $x$ for $(-x)$}: \quad& y&=-7(-x)^2\\&\textsf{Simplify}: \quad &y&=-7x^2\\\end{aligned}`

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

`\begin{aligned}&\textsf{Given}: \quad& y&=8x^2-3\\&\textsf{Replace $x$ for $(-x)$}: \quad& y&=8(-x)^2-3\\&\textsf{Simplify}: \quad &y&=8x^2-3\\\end{aligned}`

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

Answer 2

We are looking for the functions which are symmetric with respect to the y-axis.

This should be an even function with the vertex on the y-axis.

All functions of odd degree or restricted domain or range are excluded:

• Lines, cubic functions, square root or exponent functions.

Possible symmetric functions:

• Quadratic, 4th degree etc., absolute value, circles with center on the y-axis.

See the attached. Those with red cross are excluded, with green mark are correct choices.