Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t, \quad y=|t-2| $$

Short answer

The rectangular equation is a piecewise-defined function: for \(x \geq 4\), \(y = x/2 - 2\), and for \(x < 4\), \(y = 2 - x/2\). The curve is oriented upwards for \(x \geq 4\) and to the left for \(x < 4\).

Step-by-step solution

Express the parameter \(t\) in terms of \(x\)

From the first parametric equation, \(x = 2t\), we can express \(t\) in terms of \(x\) by dividing both sides by 2. We get \(t = x/2\). This expression will be useful for eliminating the parameter in the second parametric equation.

Substitute \(t\) into the second parametric equation

Substitute \(t = x/2\) from Step 1 into the second equation, \(y=|t-2|\), to get \(y = |x/2 - 2|\). This equation is in rectangular form, but the absolute value symbols need to be addressed.

Simplify the absolute value

The expression inside the absolute value can be positive or negative. For \(y = |x/2 - 2|, y = x/2 - 2\) when \(x/2 - 2 \geq 0\) or in other words when \(x \geq 4\), and \(y = -(x/2 - 2)\) or \(y = 2 - x/2\) when \(x/2 - 2 < 0\) or \(x < 4\). This gives us the piecewise-defined function for \(y\) in terms of \(x\).

Sketch the curve

Using the piecewise-defined function from Step 3, the curve can be graphed in the rectangular coordinate system. For \(x \geq 4\), \(y = x/2 - 2\), and for \(x < 4\), \(y = 2 - x/2\). The curve shifts at the point (4, 0). The orientation depends on the direction of \(t\); in this case, for larger values of \(t\), \(x\) and \(y\) both increase, so the orientation is upwards for \(x \geq 4\) and to the left for \(x < 4\).