In Exercises circleeqnfirst - circleeqnlast, graph the circle in the \(xy\)-plane. Find the center and radius.
\((x + 1)^{2} + (y + 5)^{2} = 100\)

Center \(\answer {(-1, -5)}\), radius \(\answer {10}\)

\[\graph {(x + 1)^{2} + (y + 5)^{2} = 100}\]
\((x-4)^2+(y+2)^2 = 9\)

Center \((4,-2)\), radius \(3\)
\[\graph {(x - 4)^{2} + (y + 2)^{2} = 9}\]
\(\left (x + 3\right )^{2} + \left (y - \frac {7}{13}\right )^{2} = \frac {1}{4}\)

Center \(\answer {(-3, \frac {7}{13})}\), radius \(\answer {\frac {1}{2}}\)

\[\graph {(x + 3)^{2} + (y - 7/13)^{2} = 1/4}\]
\((x - 5)^{2} + (y + 9)^{2} = (\ln (8))^{2}\)

Center \((5, -9)\), radius \(\ln (8)\)
\[\graph {(x - 5)^{2} + (y + 9)^{2} = (ln(8))^{2}}\]
\((x + e)^{2} + \left (y - \sqrt {2} \right )^{2} = \pi ^{2}\)

Center \((-e, \sqrt {2})\), radius \(\pi \)

\(\left (x - \pi \right )^{2} + \left (y - e^{2}\right )^{2} = 91^{\frac {2}{3}}\)

Center \((\pi , e^2)\), radius \(91^{1/3}\)

In Exercises ctscirclefirst - ctscirclelast, complete the square in order to put the equation into standard form. Identify the center and the radius or explain why the equation does not represent a circle.
\(x^{2} - 4x + y^{2} + 10y = -25\)

\((x - 2)^{2} + (y + 5)^{2} = 4\)
Center \((2, -5)\), radius \(r = 2\)
\(-2x^{2} - 36x - 2y^{2} - 112 = 0\)

\((x + 9)^{2} + y^{2} = 25\)
Center \((-9, 0)\), radius \(r = 5\)
\(3x^2+3y^2+24x-30y -3 =0\)

\((x+4)^2 + (y-5)^2 = 42\)
Center \((-4,5)\), radius \(r = \sqrt {42}\)
\(x^2+y^2+5x-y-1=0\)

\(\left (x + \frac {5}{2}\right )^2 + \left (y - \frac {1}{2}\right )^2 = \frac {30}{4}\)
Center \(\left ( -\frac {5}{2}, \frac {1}{2}\right )\), radius \(r = \frac {\sqrt {30}}{2}\)
\(x^{2} + x + y^{2} - \frac {6}{5}y = 1\)

\(\left (x + \frac {1}{2}\right )^{2} + \left (y - \frac {3}{5}\right )^{2} = \frac {161}{100}\)
Center \(\left (-\frac {1}{2}, \frac {3}{5}\right )\), radius \(r = \frac {\sqrt {161}}{10}\)
\(4x^{2} + 4y^{2} - 24y + 36 = 0\)

\(x^{2} + (y - 3)^{2} = 0\)
This is not a circle.
For each of the odd numbered equations given in Exercises oddcircleone - oddcircleeleven, find two or more explicit functions of \(x\) represented by each of the equations. (See Example horizontalparabolaex in Section Parabolas.)

For number oddcircleone:
  • \(f(x) = -5 + \sqrt {99-2x-x^2}\) represents the upper semicircle.
  • \(g(x) = -5 - \sqrt {99-2x-x^2}\) represents the lower semicircle.

For number oddcirclethree:

  • \(f(x) = \frac {7}{13} + \frac {1}{2} \sqrt {-4x^2-24x-35}\) represents the upper semicircle.
  • \(g(x) = \frac {7}{13} - \frac {1}{2} \sqrt {-4x^2-24x-35}\) represents the lower semicircle.

For number oddcirclefive:

  • \(f(x) = \sqrt {2} + \sqrt {\pi ^2-e^2-2ex-x^2}\) represents the upper semicircle.
  • \(g(x) = \sqrt {2} - \sqrt {\pi ^2-e^2-2ex-x^2}\) represents the lower semicircle.

For number oddcircleseven:

  • \(f(x) = -5 + \sqrt {4x-x^2}\) represents the upper semicircle.
  • \(g(x) = -5 - \sqrt {4x-x^2}\) represents the lower semicircle.

For number oddcirclenine:

  • \(f(x) = 5 + \sqrt {26-8x-x^2}\) represents the upper semicircle.
  • \(g(x) = 5 - \sqrt {26-8x-x^2}\) represents the lower semicircle.

For number oddcircleeleven:

  • \(f(x) = \frac {3}{5} + \frac {1}{5} \sqrt {34-25x-25x^2}\) represents the upper semicircle.
  • \(g(x) = \frac {3}{5} - \frac {1}{5} \sqrt {34-25x-25x^2}\) represents the lower semicircle.
In Exercises semicirclefunctionfirst - semicirclefunctionlast, graph each function by recognizing it as a semicircle.
\(f(x) = \sqrt {4-x^2}\)

\(g(x) = -\sqrt {6x-x^2}\)

\(f(x) = -\sqrt {3-2x-x^2}\)

\(g(x) = -2 + \sqrt {9-x^2}\)

In Exercises buildcirclefromgraphfirst - buildcirclefromgraphlast, find an equation for the circle or semicircle whose graph is given.

[Picture]

\(\answer {(x-1)^2+y^2=9}\)

[Picture]

\((x-4)^2+(y-4)^2=16\)
[Picture]

\(\answer {y = 4-\sqrt {16-x^2}}\)

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\(y = \sqrt {8x-x^2}\)
In Exercises buildcirclefirst - buildcirclelast, find the standard equation of the circle which satisfies the given criteria.
center \((3, 5)\), passes through \((-1, -2)\)

\((x - 3)^{2} + (y - 5)^{2} = 65\)
center \((3, 6)\), passes through \((-1, 4)\)

\((x-3)^2+(y-6)^2 = 20\)
endpoints of a diameter: \((3,6)\) and \((-1,4)\)

\((x-1)^2 + (y-5)^2 = 5\)
endpoints of a diameter: \(\left ( \frac {1}{2}, 4\right )\), \(\left (\frac {3}{2}, -1\right )\)

\((x-1)^2 + \left (y - \frac {3}{2}\right )^2 = \frac {13}{2}\)
The Giant Wheel at Cedar Point is a circle with diameter 128 feet which sits on an 8 foot tall platform making its overall height is 136 feet. Find an equation for the wheel assuming that its center lies on the \(y\)-axis and that the ground is the \(x\)-axis.

\(x^{2} + (y - 72)^{2} = 4096\)
Verify that the following points lie on the Unit Circle:

\((\pm 1, 0)\), \((0, \pm 1)\), \(\left (\pm \frac {\sqrt {2}}{2}, \pm \frac {\sqrt {2}}{2}\right )\), \(\left (\pm \frac {1}{2}, \pm \frac {\sqrt {3}}{2}\right )\) and \(\left (\pm \frac {\sqrt {3}}{2}, \pm \frac {1}{2}\right )\)

Discuss with your classmates how to obtain the alternate standard equation of a circle, Equation standardcirclealternate, from the equation of the Unit Circle, \(x^2+y^2=1\) using the transformations discussed in Section Transformations. (Thus every circle is just a few transformations away from the Unit Circle.)
Find a one-to-one function whose graph is half of a circle.

Think piecewise …