Chapter 8

(a) The parametric equations \(x=f(t)\) and \(y=g(t)\) give the coordinates of a point \((x, y)=(f(t), g(t))\) for appropriate values of \(t .\) The variable \(t\) is called a _______. (b) Suppose that the parametric equations \(x=t, y=t^{2}\) \(t \geq 0,\) model the position of a moving object at time \(t\) When \(t=0,\) the object is at \((\square, \square),\) and when \(t=1\) the object is at \((\square, \square)\). (c) If we eliminate the parameter in part (b), we get the equation \(y=\) _______. We see from this equation that the path of the moving object is a _______.

A complex number \(z=a+b i\) has two parts: \(a\) is the _____ part, and \(b\) is the _____. To graph \(a+b i\) we graph the ordered pair \((\square, \square)\) in the complex plane.

Let \(z=a+b i\) (a) The modulus of \(z\) is \(r=\) _____ and an argument of \(z\) is an angle \(\theta\) satisfying \(\tan \theta=\)_____. (b) We can express \(z\) in polar form as \(z=\)_____ where \(r\) is the modulus of \(z\) and \(\theta\) is the argument of \(z\).

Let \(P\) be a point in the plane. (a) If \(P\) has polar coordinates \((r, \theta)\) then it has rectangular coordinates \((x, y)\) where \(x=\) ____________ and $$y=$$ ____________. (b) If \(P\) has rectangular coordinates \((x, y)\) then it has polar coordinates \((r, \theta)\) where \(r^{2}=\) ____________ and $$\tan \theta=$$ ____________.

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 t, \quad y=t+6$$

Plot the point that has the given polar coordinates. $$(4, \pi / 4)$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=6 t-4, \quad y=3 t, \quad t \geq 0$$

Plot the point that has the given polar coordinates. $$(1,0)$$

Graph the complex number and find its modulus. $$4 i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=t^{2}, \quad y=t-2, \quad 2 \leq t \leq 4$$

Plot the point that has the given polar coordinates. $$(6,-7 \pi / 6)$$

Graph the complex number and find its modulus. $$-3 i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 t+1, \quad y=\left(t+\frac{1}{2}\right)^{2}$$

Plot the point that has the given polar coordinates. $$(3,-2 \pi / 3)$$

Graph the complex number and find its modulus. $$-2$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sqrt{t}, \quad y=1-t$$

Plot the point that has the given polar coordinates. $$(-2,4 \pi / 3)$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=t^{2}, \quad y=t^{4}+1$$

Plot the point that has the given polar coordinates. $$(-5,-17 \pi / 6)$$

Graph the complex number and find its modulus. $$5+2 i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\frac{1}{t}, \quad y=t+1$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r=2-\sin \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$(3, \pi / 2)$$

Graph the complex number and find its modulus. $$7-3 i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=t+1, \quad y=\frac{t}{t+1}$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r=4+8 \cos \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$(2,3 \pi / 4)$$

Graph the complex number and find its modulus. $$\sqrt{3}+i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=4 t^{2}, \quad y=8 t^{3}$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r=3 \sec \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$(-1,7 \pi / 6)$$

Graph the complex number and find its modulus. $$-1-\frac{\sqrt{3}}{3} i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=|t|, \quad y=|1-| t||$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$(-2,-\pi / 3)$$

Graph the complex number and find its modulus. $$\frac{3+4 i}{5}$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 \sin t, \quad y=2 \cos t, \quad 0 \leq t \leq \pi$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r=\frac{4}{3-2 \sin \theta}$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$(-5,0)$$

Graph the complex number and find its modulus. $$\frac{-\sqrt{2}+i \sqrt{2}}{2}$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r=\frac{5}{1+3 \cos \theta}$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$(3,1)$$

Sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$z=1+i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sin ^{2} t, \quad y=\sin ^{4} t$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r^{2}=4 \cos 2 \theta$$

Sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$z=-1+i \sqrt{3}$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sin ^{2} t, \quad y=\cos t$$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r^{2}=9 \sin \theta$$

Sketch the complex number \(z\) and its complex conjugate \(z\) on the same complex plane. $$z=8+2 i$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos t, \quad y=\cos 2 t$$