Question

find the equation of the inverse. graph both equations on the same coordinate plane.

1. ( y = 2x )
2. ( y = x^2 )

Explanation:

For \(y=2x\):

Step1: Swap \(x\) and \(y\)

\(x = 2y\)

Step2: Solve for \(y\)

\(y = \frac{x}{2}\)

Step3: Graph key points for \(y=2x\)

Points: \((0,0)\), \((1,2)\), \((-1,-2)\); draw a straight line through them.

Step4: Graph key points for \(y=\frac{x}{2}\)

Points: \((0,0)\), \((2,1)\), \((-2,-1)\); draw a straight line through them.

For \(y=x^2\):

Step1: Swap \(x\) and \(y\)

\(x = y^2\)

Step2: Solve for \(y\)

\(y = \pm\sqrt{x}\) (domain \(x\geq0\))

Step3: Graph key points for \(y=x^2\)

Points: \((0,0)\), \((1,1)\), \((-1,1)\), \((2,4)\), \((-2,4)\); draw a parabola opening upward.

Step4: Graph key points for \(y=\pm\sqrt{x}\)

For \(y=\sqrt{x}\): \((0,0)\), \((1,1)\), \((4,2)\); for \(y=-\sqrt{x}\): \((0,0)\), \((1,-1)\), \((4,-2)\); draw two curves opening right.

Answer:

1. Inverse of \(y=2x\): \(\boldsymbol{y=\frac{x}{2}}\)

• The graph of \(y=2x\) is a straight line with slope 2 passing through the origin; \(y=\frac{x}{2}\) is a straight line with slope \(\frac{1}{2}\) passing through the origin, symmetric to \(y=2x\) across \(y=x\).

1. Inverse of \(y=x^2\): \(\boldsymbol{y=\pm\sqrt{x} \ (x\geq0)}\)

• The graph of \(y=x^2\) is an upward-opening parabola with vertex at the origin; its inverse is a pair of right-opening curves symmetric across the x-axis, symmetric to \(y=x^2\) across \(y=x\).