Section 1
The Circle and the Parabola
Solve each equation.$|3 x-4|=11$
Solve each equation.$\left|\frac{4-3 x}{5}\right|=12$
Solve each equation. $|3 x+4|=|5 x-2|$
Solve each equation.$|6-4 x|=|x+2|$
Fill in the blanks.A ________ is the set of all points in a ________ that are a fixed distance from a given point.
Fill in the blanks.The fixed distance in Exercise 5 is called the ________ of the circle, and the point is called its ________
Fill in the blanks.If ________ in the equation $x^2+y^2=r^2$, no circle exists.
Fill in the blanks.The graph of $y=a x^2 \quad(a>0)$ is a ________ with vertex at the ________ that opens ________
Fill in the blanks.The graph of $x=a(y-2)^2+3 \quad(a>0)$ is a ________with vertex at ________ that opens to the ________
Fill in the blanks.The graph of $x=a(y-1)^2-3 \quad(a<0)$ is a ________with vertex at ________ that opens to the ________
Graph each equation.$x^2+y^2=9$(graph can't copy)
Graph each equation.$x^2+y^2=16$(graph can't copy)
Graph each equation.$(x-2)^2+y^2=9$(graph can't copy)
Graph each equation.$x^2+(y-3)^2=4$(graph can't copy)
Graph each equation. $(x-2)^2+(y-4)^2=4$(graph can't copy)
Graph each equation. $(x-3)^2+(y-2)^2=4$(graph can't copy)
Graph each equation.$(x+3)^2+(y-1)^2=16$(graph can't copy)
Graph each equation. $(x-1)^2+(y+4)^2=9$(graph can't copy)
Graph each equation. $x^2+(y+3)^2=1$(graph can't copy)
Graph each equation.$(x+4)^2+y^2=1$(graph can't copy)
Use a graphing calculator to graph each equation.$3 x^2+3 y^2=16$
Use a graphing calculator to graph each equation.$2 x^2+2 y^2=9$
Use a graphing calculator to graph each equation.$(x+1)^2+y^2=16$
Use a graphing calculator to graph each equation.$x^2+(y-2)^2=4$
Write the equation of the circle with the following properties.Center at origin; radius 1
Write the equation of the circle with the following properties.Center at origin; radius 4
Write the equation of the circle with the following properties. Center at $(6,8)$; radius 5
Write the equation of the circle with the following properties.Center at $(5,3)$; radius 2
Write the equation of the circle with the following properties.Center at $(-2,6)$; radius 12
Write the equation of the circle with the following properties.Center at $(5,-4)$; radius 6
Write the equation of the circle with the following properties.Center at the origin; diameter of $2 \sqrt{2}$
Write the equation of the circle with the following properties.Center at the origin; diameter of $8 \sqrt{3}$
Graph each circle. Give the coordinates of the center:$x^2+y^2+2 x-8=0$(graph can't copy)
Graph each circle. Give the coordinates of the center:$x^2+y^2-4 y=12$(graph can't copy)
Graph each circle. Give the coordinates of the center:$9 x^2+9 y^2-12 y=5$(graph can't copy)
Graph each circle. Give the coordinates of the center: $4 x^2+4 y^2+4 y=15$(graph can't copy)
Graph each circle. Give the coordinates of the center:$x^2+y^2-2 x+4 y=-1$(graph can't copy)
Graph each circle. Give the coordinates of the center:$x^2+y^2+4 x+2 y=4$(graph can't copy)
Graph each circle. Give the coordinates of the center: $x^2+y^2+6 x-4 y=-12$(graph can't copy)
Graph each circle. Give the coordinates of the center:$x^2+y^2+8 x+2 y=-13$(graph can't copy)
Find the vertex of each parabola and graph it.$x=y^2$(graph can't copy)
Find the vertex of each parabola and graph it.$x=-y^2+1$(graph can't copy)
Find the vertex of each parabola and graph it. $x=-\frac{1}{4} y^2$(graph can't copy)
Find the vertex of each parabola and graph it.$x=4 y^2$(graph can't copy)
Find the vertex of each parabola and graph it. $y=x^2+4 x+5$(graph can't copy)
Find the vertex of each parabola and graph it. $y=-x^2-2 x+3$(graph can't copy)
Find the vertex of each parabola and graph it.$y=-x^2-x+1$(graph can't copy)
Find the vertex of each parabola and graph it.$x=\frac{1}{2} y^2+2 y$(graph can't copy)
Find the vertex of each parabola and graph it.$y^2+4 x-6 y=-1$(graph can't copy)
Find the vertex of each parabola and graph it.$x^2-2 y-2 x=-7$(graph can't copy)
Find the vertex of each parabola and graph it. $y=2(x-1)^2+3$(graph can't copy)
Find the vertex of each parabola and graph it. $y=-2(x+1)^2+2$(graph can't copy)
Use a graphing calculator to graph each equation.$x=2 y^2$
Use a graphing calculator to graph each equation. $x=y^2-4$
Use a graphing calculator to graph each equation.$x^2-2 x+y=6$
Use a graphing calculator to graph each equation.$x=-2(y-1)^2+2$
For design purposes, the large gear is the circle $x^2+y^2=16$. The smaller gear is a circle centered at $(7,0)$ and tangent to the larger circle. Find the equation of the smaller gear.
The following walkway is bounded by the two circles $x^2+y^2=2,500$ and $(x-10)^2+y^2=900$, measured in feet. Find the largest and the smallest width of the walkway.
Radio stations applying for licensing may not use the same frequency if their broadcast areas overlap. One station's coverage is bounded by $x^2+y^2-8 x-20 y+16=0$, and the other's by $x^2+y^2+2 x+4 y-11=0$. May they be licensed for the same frequency?
Engineers want to join two sections of highway with a curve that is one-quarter of a circle. The equation of the circle is $x^2+y^2-16 x-20 y+155=0$, where distances are measured in kilometers. Find the locations (relative to the center of town) of the intersections of the highway with State and with Main.
The cannonball in the illustration follows the parabolic trajectory $y=30 x-x^2$. Where does it land?
In Exercise 61, how high does the cannonball get?
If the orbit of a comet is given by the equation $2 y^2-9 x=18$, how far is it from the sun at the vertex of the orbit? Distances are measured in astronomical units (AU).
The cross section of the satellite antenna is a parabola given by the equation $y=\frac{1}{16} x^2$, with distances measured in feet. If the dish is 8 feet wide, how deep is it?
Explain how to decide from its equation whether the graph of a parabola opens up, down, right, or left.
From the equation of a circle, explain how to determine the radius and the coordinates of the center.
From the values of $a, h$, and $k$, explain how to determine the number of $x$-intercepts of the graph of $y=a(x-h)^2+k$.
Under what conditions will the graph of $x=a(y-k)^2+h$ have no $y$-intercepts?