Question

determine whether the graph of each of the following equations is symmetric with respect to the x-axis,y-axis,origin the line y=x or none of these.

a)y= -8x

b)x^2+y^2=4

c)y=-|x|

Answer 1

(a).

The function is y = - 8x.

Test y = - 8x for symmetry with respect to x-axes :

Replace y with - y.

(- y) = - 8x

y = 8x

This is not an equivalent equation.

Test y = - 8x for symmetry with respect to y-axes :

Replace x with - x.

y = - 8(- x)

y = 8x

This is not an equivalent equation.

Test y = - 8x for symmetry with respect to origin :

Replace y with - y and x with - x.

(- y) = - 8(- x)

- y = 8x

y = - 8x

This is an equivalent equation, so the graph of y = - 8x for symmetry with respect to origin.

Answer 2

(b).

The function is x² + y² = 4.

Test x² + y² = 4 for symmetry with respect to x-axes :

Replace y with - y.

x² + (- y)² = 4

x² + y² = 4

This is an equivalent equation, so the graph of x² + y² = 4 for symmetry with respect to x-axis.

Test x² + y² = 4 for symmetry with respect to y-axes :

Replace x with - x.

(- x)² + y² = 4

x² + y² = 4

This is an equivalent equation, so the graph of x² + y² = 4 for symmetry with respect to y-axis.

Test x² + y² = 4 for symmetry with respect to origin :

Replace y with - y and x with - x.

(- x)² + (- y)² = 4

x² + y² = 4

This is an equivalent equation, so the graph of x² + y² = 4 for symmetry with respect to origin.

Answer 3

(c).

The function is y = - |x|.

Test y = - |x| for symmetry with respect to x-axes :

Replace y with - y.

(- y) = - | x |

y = | x |

This is not an equivalent equation.

Test y = - |x| for symmetry with respect to y-axes :

Replace x with - x.

y = - |(- x)|

y = - |- 1| | x |

y = - | x |

This is an equivalent equation, so the graph of y = - |x| for symmetry with respect to y-axis.

Test y = - |x| for symmetry with respect to origin :

Replace y with - y and x with - x.

(- y) = - |(- x)|

- y = - |-1||x|

- y = - |x|

y = |x|

This is not an equivalent equation.