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Parametric Equations

  1. A bug moves linearly with constant speed across my graph paper. I first notice the bug when it is at \((3,4)\). It reaches \((9, 8)\) after two seconds and \((15, 12)\) after four seconds.

    1. Predict the position of the bug after 6 seconds, after 9 seconds; after \(t\) seconds.
    2. Is there a time when the bug is equidistant from the x and y-axes? If so, where is it?

  2. The x and y-coordinates of points that depend on \(t\) are given by the equations shown. Use your graph paper to plot points corresponding to \(t=-1\), \(0\), and \(2\). These points should appear to be collinear. Convince yourself that this is the case and calculate the slope of this line. How is the slope of a line determined from its parametric equations?

  3. Find parametric equations to describe the line that goes through the points \(A = (5, -3)\) and \(B = (7, 1)\). Find at least two parametric equations that work. What is the difference between them?

  4. Caught in a nightmare, Blair is moving along the line \(y = 3x + 2\). At midnight, Blair's position is \((1, 5)\) and the x-coordinate increasing by 4 units every hour. Write parametric equations that describe Blair's position \(t\) hours after midnight. What was Blair's position at 10:15 pm when the nightmare started? Find Blair's speed, in units per hour.

  5. The parametric equations \(x=-2-3t\) and \(y=6+4t\) describe the position of a particle, in meters and seconds. How does the particle's position change each second? Each minute? What is the speed of the particle, in meters per second? Write parametric equations that describe the particle's position, using meters and minutes as units.

  6. Find parametric equations that describe the following lines:

    1. Through \((3, 1)\) and \((7, 3)\)
    2. Through \((7,-1)\) and \((7, 3)\)

  7. A bug is moving along the line \(3x + 4y = 12\) with constant speed 5 units per second. The bug crosses the x-axis when \(t = 0\) seconds. It crosses the y-axis later. When? Where is the bug when \(t = 2\)? when \(t=-1\)? When \(t = 1.5\)? What does a negative t-value mean?

  8. Find the coordinates for the point that is three fifths of the way from \((4, 0)\) to \((0, 3)\). Start by writing parametric equations to describe the line joining those two points.

  9. The position of a bug is described by the parametric equation \((x, y) = (2-12t, 1+5t)\). Explain why the speed of the bug is \(13\) cm/sec. Change the equation to obtain the description of a bug moving along the same line with speed \(26\) cm/second.