Determine whether the graph of the equation is symmetric with respect to the $y$-axis, the $x$-axis, the origin, more than one of these, or none of these.
$y^2=x^2+17$
Select all that apply.
x-axis
y-axis
origin
none of these
Answer (2)
Replace y with − y to test for x -axis symmetry: y 2 = x 2 + 17 .
Replace x with − x to test for y -axis symmetry: y 2 = x 2 + 17 .
Replace x with − x and y with − y to test for origin symmetry: y 2 = x 2 + 17 .
The graph is symmetric with respect to the x -axis, y -axis, and the origin: x -axis, y -axis, origin
Explanation
Understanding the Problem We are given the equation y 2 = x 2 + 17 and we need to determine its symmetry with respect to the x -axis, y -axis, and the origin.
Testing for x-axis Symmetry To test for symmetry with respect to the x -axis, we replace y with − y in the equation. If the equation remains unchanged, then the graph is symmetric with respect to the x -axis. Substituting − y for y , we get ( − y ) 2 = x 2 + 17 , which simplifies to y 2 = x 2 + 17 . Since this is the same as the original equation, the graph is symmetric with respect to the x -axis.
Testing for y-axis Symmetry To test for symmetry with respect to the y -axis, we replace x with − x in the equation. If the equation remains unchanged, then the graph is symmetric with respect to the y -axis. Substituting − x for x , we get y 2 = ( − x ) 2 + 17 , which simplifies to y 2 = x 2 + 17 . Since this is the same as the original equation, the graph is symmetric with respect to the y -axis.
Testing for Origin Symmetry To test for symmetry with respect to the origin, we replace x with − x and y with − y in the equation. If the equation remains unchanged, then the graph is symmetric with respect to the origin. Substituting − x for x and − y for y , we get ( − y ) 2 = ( − x ) 2 + 17 , which simplifies to y 2 = x 2 + 17 . Since this is the same as the original equation, the graph is symmetric with respect to the origin.
Conclusion Since the graph is symmetric with respect to the x -axis, the y -axis, and the origin, we select all three options.
Examples
Symmetry is a fundamental concept in mathematics and physics. For example, the trajectory of a projectile in a uniform gravitational field, neglecting air resistance, is symmetric about its highest point. Similarly, many physical laws, such as those governing electromagnetism, exhibit symmetry under certain transformations, which simplifies their analysis and leads to conservation laws.
The graph of the equation y 2 = x 2 + 17 is symmetric with respect to the x-axis, y-axis, and origin. Therefore, the selected options are: x-axis, y-axis, and origin.
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