Section 1
Circles and Parabolas
In Exercises $1-6,$ find the standard form of the equation of the circle with the given characteristics. Center at origin; radius: $\sqrt{18}$
In Exercises $1-6,$ find the standard form of the equation of the circle with the given characteristics. Center at origin; radius: 4$\sqrt{2}$
In Exercises $1-6,$ find the standard form of the equation of the circle with the given characteristics. Center: $(3,7) ; \quad$ point on circle: $(1,0)$
In Exercises $1-6,$ find the standard form of the equation of the circle with the given characteristics. Center: $(6,-3) ;$ point on circle: $(-2,4)$
In Exercises $1-6,$ find the standard form of the equation of the circle with the given characteristics. Center: $(-3,-1) ; \quad$ diameter: 2$\sqrt{7}$
In Exercises $1-6,$ find the standard form of the equation of the circle with the given characteristics. Center: $(5,-6) ; \quad$ diameter: 4$\sqrt{3}$
In Exercises $7-12,$ identify the center and radius of the circle. $$x^{2}+y^{2}=49$$
In Exercises $7-12,$ identify the center and radius of the circle. $$x^{2}+y^{2}=1$$
In Exercises $7-12,$ identify the center and radius of the circle. $$(x+2)^{2}+(y-7)^{2}=16$$
In Exercises $7-12,$ identify the center and radius of the circle. $$(x+9)^{2}+(y+1)^{2}=36$$
In Exercises $7-12,$ identify the center and radius of the circle. $$(x-1)^{2}+y^{2}=15$$
In Exercises $7-12,$ identify the center and radius of the circle. $$x^{2}+(y+12)^{2}=24$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$\frac{1}{4} x^{2}+\frac{1}{4} y^{2}=1$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$\frac{1}{9} x^{2}+\frac{1}{9} y^{2}=1$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$\frac{4}{3} x^{2}+\frac{4}{3} y^{2}=1$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$\frac{9}{2} x^{2}+\frac{9}{2} y^{2}=1$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$x^{2}+y^{2}-2 x+6 y+9=0$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$x^{2}+y^{2}-10 x-6 y+25=0$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$4 x^{2}+4 y^{2}+12 x-24 y+41=0$$
In Exercises $13-20,$ write the equation of the circle in standard form. Then identify its center and radius. $$9 x^{2}+9 y^{2}+54 x-36 y+17=0$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}=16-y^{2}$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$y^{2}=81-x^{2}$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}+4 x+y^{2}+4 y-1=0$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}-6 x+y^{2}+6 y+14=0$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}-14 x+y^{2}+8 y+40=0$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}+6 x+y^{2}-12 y+41=0$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}+2 x+y^{2}-35=0$$
In Exercises $21-28$ , sketch the circle. Identify its center and radius. $$x^{2}+y^{2}+10 y+9=0$$
In Exercises $29-34,$ find the $x$ - and $y$ -intercepts of the graph of the circle. $$(x-2)^{2}+(y+3)^{2}=9$$
In Exercises $29-34,$ find the $x$ - and $y$ -intercepts of the graph of the circle. $$(x+5)^{2}+(y-4)^{2}=25$$
In Exercises $29-34,$ find the $x$ - and $y$ -intercepts of the graph of the circle. $$x^{2}-2 x+y^{2}-6 y-27=0$$
In Exercises $29-34,$ find the $x$ - and $y$ -intercepts of the graph of the circle. $$x^{2}+8 x+y^{2}+2 y+9=0$$
In Exercises $29-34,$ find the $x$ - and $y$ -intercepts of the graph of the circle. $$(x-6)^{2}+(y+3)^{2}=16$$
In Exercises $29-34,$ find the $x$ - and $y$ -intercepts of the graph of the circle. $$(x+7)^{2}+(y-8)^{2}=4$$
Earthquake An earthquake was felt up to 81 miles from its epicenter. You were located 60 miles west and 45 miles south of the epicenter. (a) Let the epicenter be at the point $(0,0) .$ Find the standard equation that describes the outer boundary of the earthquake.(b) Would you have felt the earthquake?(c) Verify your answer to part (b) by graphing the equation of the outer boundary of the earthquake and plotting your location. How far were you from the outer boundary of the earthquake?
Landscaper $A$ landscaper has installed a circular sprinkler system that covers an area of 1800 square feet.(a) Find the radius of the region covered by the sprinkler system. Round your answer to three decimal places.(b) If the landscaper wants to cover an area of 2400 square feet, how much longer does the radius need to be?
In Exercises $37-42,$ match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f). $$y^{2}=-4 x$$
In Exercises $37-42,$ match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f). $$x^{2}=2 y$$
In Exercises $37-42,$ match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f). $$x^{2}=-8 y$$
In Exercises $37-42,$ match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f). $$y^{2}=-12 x$$
In Exercises $37-42,$ match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f). $$(y-1)^{2}=4(x-3)$$
In Exercises $37-42,$ match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f). $$(x+3)^{2}=-2(y-1)$$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: $\left(0,-\frac{3}{2}\right)$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: $\left(\frac{5}{2}, 0\right)$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: $(-2,0)$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: $(0,1)$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: $y=-1$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: $y=3$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: $x=2$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: $x=-3$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Horizontal axis and passes through the point $(4,6)$
In Exercises $43-54$ , find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Vertical axis and passes through the point $(-3,-3)$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y=\frac{1}{2} x^{2}$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y=-4 x^{2}$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}=-6 x$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}=3 x$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$x^{2}+8 y=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$x+y^{2}=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$(x+1)^{2}+8(y+3)=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$(x-5)+(y+4)^{2}=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}+6 y+8 x+25=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}-4 y-4 x=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$\left(x+\frac{3}{2}\right)^{2}=4(y-2)$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$\left(x+\frac{1}{2}\right)^{2}=4(y-1)$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y=\frac{1}{4}\left(x^{2}-2 x+5\right)$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$x=\frac{1}{4}\left(y^{2}+2 y+33\right)$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$x^{2}+4 x+6 y-2=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$x^{2}-2 x+8 y+9=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}+x+y=0$$
In Exercises $55-72$ , find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}-4 x-4=0$$
In Exercises $73-82$ , find the standard form of the equation of the parabola with the given characteristics.
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Vertex: $(-2,0) ;$ focus: $\left(-\frac{3}{2}, 0\right)$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Vertex: $(3,-3) ;$ focus: $\left(3,-\frac{9}{4}\right)$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Vertex: $(5,2) ;$ focus: $(3,2)$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Vertex: $(-1,2) ;$ focus: $(-1,0)$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Vertex: $(0,4) ;$ directrix: $y=2$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Vertex: $(-2,1) ;$ directrix: $x=1$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Focus: $(2,2) ;$ directrix: $x=-2$
In Exercises $73-82,$ find the standard form of the equation of the parabola with the given characteristics. Focus: $(0,0) ;$ directrix: $y=4$
In Exercises 83 and $84,$ the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both in the same viewing window. Determine the coordinates of the point of tangency.$$y^{2}-8 x=0 \quad x-y+2=0$$
In Exercises 83 and $84,$ the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both in the same viewing window. Determine the coordinates of the point of tangency.$$x^{2}+12 y=0 \quad x+y-3=0$$
In Exercises $85-88,$ find an equation of the tangent line to the parabola at the given point and find the $x$ -intercept of the line. $$x^{2}=2 y,(4,8)$$
In Exercises $85-88,$ find an equation of the tangent line to the parabola at the given point and find the $x$ -intercept of the line. $$x^{2}=2 y,\left(-3, \frac{9}{2}\right)$$
In Exercises $85-88,$ find an equation of the tangent line to the parabola at the given point and find the $x$ -intercept of the line. $$y=-2 x^{2},(-1,-2)$$
In Exercises $85-88,$ find an equation of the tangent line to the parabola at the given point and find the $x$ -intercept of the line. $$y=-2 x^{2},(2,-8)$$
Revenue The revenue $R$ (in dollars) generated by the sale of $x 32$ -inch televisions is modeled by $R=375 x-\frac{3}{2} x^{2}$ . Use a graphing utility to graph the function and approximate the sales that will maximize revenue.
Beam Deflection A simply supported beam is 64 feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. (a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.)(b) How far from the center of the beam is the deflection equal to $\frac{1}{2}$ inch?
Automobile Headlight The filament of an automobile headlight is at the focus of a parabolic reflector, which sends light out in a straight beam (see figure). (a) The filament of the headlight is 1.5 inches from the vertex. Find an equation for the cross section of the reflector.(b) The reflector is 8 inches wide. Find the depth of the reflector.
Solar Cooker You want to make a solar hot dog cooker using aluminum foil-lined cardboard, shaped as a parabolic trough. The figure shows how to suspend the hot dog with a wire through the foci of the ends of the parabolic trough. The parabolic end pieces are 12 inches wide and 4 inches deep. How far from the bottom of the trough should the wire be inserted?
Suspension Bridge Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the road- way. Label the coordinates of the known points.(b) Write an equation that models the cables.(c) Complete the table by finding the height y of the suspension cables over the roadway at a distance of $x$ meters from the center of the bridge. $$\begin{array}{|c|c|c|c|c|c|}\hline x & {0} & {200} & {400} & {500} & {600} \\ \hline y & {} & {} & {} & {} \\ \hline\end{array}$$
Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.)(b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?
Highway Design Highway engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highway (see figure). Find an equation of the parabola.
Satellite Orbit A satellite in a 100 -mile-high circular orbit around Earth has a velocity of approximately $17,500$ miles per hour. If this velocity is multiplied by $\sqrt{2},$ the satellite will have the minimum velocity necessary to escape Earth's gravity, and it will follow a parabolic path with the center of Earth as the focus (see figure).(a) Find the escape velocity of the satellite.(b) Find an equation of its path (assume the radius of Earth is 4000 miles).
Path of a Projectile The path of a softball is modeled by $$-12.5(y-7.125)=(x-6.25)^{2}$$ The coordinates $x$ and $y$ are measured in feet, with $x=0$ corresponding to the position from which the ball was thrown.(a) Use a graphing utility to graph the trajectory of the softball.(b) Use the zoom and trace features of the graphing utility to approximate the highest point the ball reaches and the distance the ball travels.
Projectile Motion Consider the path of a projectile projected horizontally with a velocity of $v$ feet per second at a height of $s$ feet, where the model for the path is $x^{2}=-\frac{1}{16} v^{2}(y-s) .$ In this model, air resistance is disregarded, $y$ is the height (in feet) of the projectile, and $x$ is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 75 -foot tower with a velocity of 32 feet per second. (a) Find the equation of the parabolic path.(b) How far does the ball travel horizontally before striking the ground?
In Exercises $99-102,$ find an equation of the tangent line to the circle at the indicated point. Recall from geometry that the tangent line to a circle is perpendicular to the radius of the circle at the point of tangency. $$x^{2}+y^{2}=25 \quad(3,-4)$$
In Exercises $99-102,$ find an equation of the tangent line to the circle at the indicated point. Recall from geometry that the tangent line to a circle is perpendicular to the radius of the circle at the point of tangency. $$x^{2}+y^{2}=169 \quad(-5,12)$$
In Exercises $99-102,$ find an equation of the tangent line to the circle at the indicated point. Recall from geometry that the tangent line to a circle is perpendicular to the radius of the circle at the point of tangency. $$x^{2}+y^{2}=12$$
In Exercises $99-102,$ find an equation of the tangent line to the circle at the indicated point. Recall from geometry that the tangent line to a circle is perpendicular to the radius of the circle at the point of tangency. $$x^{2}+y^{2}=24 \quad(-2 \sqrt{5}, 2)$$
True or False? In Exercises $103-108$ , determine whether the statement is true or false. Justify your answer. The equation $x^{2}+(y+5)^{2}=25$ represents a circle with its center at the origin and a radius of $5 .$
True or False? In Exercises $103-108$ , determine whether the statement is true or false. Justify your answer. The graph of the equation $x^{2}+y^{2}=r^{2}$ will have $x$ -intercepts $( \pm r, 0)$ and $y$ -intercepts $(0, \pm r) .$
True or False? In Exercises $103-108$ , determine whether the statement is true or false. Justify your answer. A circle is a degenerate conic.
True or False? In Exercises $103-108$ , determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.
True or False? In Exercises $103-108$ , determine whether the statement is true or false. Justify your answer. The point which lies on the graph of a parabola closest to its focus is the vertex of the parabola.
True or False? In Exercises $103-108$ , determine whether the statement is true or false. Justify your answer. The directrix of the parabola $x^{2}=y$ intersects, or is tangent to, the graph of the parabola at its vertex, $(0,0)$ .
Writing Cross sections of television antenna dishes are parabolic in shape (see figure). Write a paragraphdescribing why these dishes are parabolic. Include a graphical representation of your description.
Think About It The equation $x^{2}+y^{2}=0$ is a degenerate conic. Sketch the graph of this equation and identify the degenerate conic. Describe the intersection of the plane with the double-napped cone for this particular conic.
Think About It In Exercises 111 and $112,$ change the equation so that its graph matches the description. $(y-3)^{2}=6(x+1) ;$ upper half of parabola
Think About It In Exercises 111 and $112,$ change the equation so that its graph matches the description. $(y+1)^{2}=2(x-2) ;$ lower half of parabola
In Exercises $113-116$ , use a graphing utility to approximate any relative minimum or maximum values of the function. $$f(x)=3 x^{3}-4 x+2$$
In Exercises $113-116$ , use a graphing utility to approximate any relative minimum or maximum values of the function. $$f(x)=2 x^{2}+3 x$$
In Exercises $113-116$ , use a graphing utility to approximate any relative minimum or maximum values of the function. $$f(x)=x^{4}+2 x+2$$
In Exercises $113-116$ , use a graphing utility to approximate any relative minimum or maximum values of the function. $$f(x)=x^{5}-3 x-1$$