Question

Plot each point. Then plot the point that is symmetric to it with respect to (a) the \(x\) -axis; (b) the y-axis; (c) the origin. $$ (-1,-1) $$

Short answer

The symmetric points are \((-1, 1)\), \(1, -1)\), and \(1, 1)\).

Step-by-step solution

Plot the Original Point

Plot the point \((-1, -1)\) on the coordinate plane. This point is located 1 unit left of the origin and 1 unit down.

Plot Symmetric Point with Respect to the x-axis

To find the point symmetric to \((-1, -1)\) with respect to the \({x}\)-axis, reflect the point over the \({x}\)-axis. The x-coordinate remains the same, while the y-coordinate changes sign. So, the symmetric point is \((-1, 1)\).

Plot Symmetric Point with Respect to the y-axis

To find the point symmetric to \((-1, -1)\) with respect to the \({y}\)-axis, reflect the point over the \({y}\)-axis. The y-coordinate remains the same, while the x-coordinate changes sign. So, the symmetric point is \(1, -1)\).

Plot Symmetric Point with Respect to the Origin

To find the point symmetric to \((-1, -1)\) with respect to the origin, both the x-coordinate and the y-coordinate change signs. The symmetric point is \(1, 1)\).

Key concepts

Plotting Points

In coordinate geometry, plotting points is about locating a point on the coordinate plane using its coordinates, \((x, y)\). The coordinate plane consists of two perpendicular lines, the \({x}\text{-axis}\text{ and }{y}\text{-axis}\text{, that intersect at the origin} (0,0).\) To plot a point like \((-1, -1)\), move 1 unit left from the origin along the \({x}\text{-axis, and then move 1 unit down along the }{y}\text{-axis}\text{.\) Place a dot where these movements intersect, and you have plotted your point.}

x-axis symmetry

Symmetry with respect to the \({x}\text{-axis occurs when a point} (x, y)\) is reflected over the \({x}\text{-axis, resulting in the symmetric point }(x, -y)\text{.\) To find the symmetric point of \((-1, -1)\) with respect to the \({x}\text{-axis}\text{, we keep the }{x}\text{-coordinate the same and change the sign of the }{y}\text{-coordinate.\) Hence, the symmetric point is \((-1, 1)\). This means the point is now located 1 unit left of the origin and 1 unit up along the \({y}\text{-axis.\)

y-axis symmetry

Symmetry about the \({y}\text{-axis involves reflecting a point }(x, y)\text{ over this axis to get the point }(-x, y)\text{.\) For the point \((-1, -1)\), to find its \({y}\text{-axis symmetric counterpart,\) we change the sign of the \({x}\text{-coordinate while leaving the }{y}\text{-coordinate unchanged.\) Therefore, the symmetric point is \((1, -1)\text{.\) This new point is located 1 unit right of the origin and 1 unit down along the \({y}\text{-axis.\)

origin symmetry

Symmetry with respect to the origin means reflecting a point \((x, y)\text{ over the origin, resulting in the point }(-x, -y)\text{.\) For the point \((-1, -1)\), reflection over the origin changes the signs of both \({x}\text{-coordinate and }{y}\text{-coordinate.\) Thus, the symmetric point is \((1, 1)\text{.\) This places the new point 1 unit right of the origin and 1 unit up, combining the effects of \({x}\text{-axis and }{y}\text{-axis symmetries.\)