Chapter 8

\(9 x^{2}+y^{2}=36\)

\(4 x^{2}-9 y^{2}=16\)

\(y^{2}-4 x^{2}=9\)

\(9 x^{2}-4 y^{2}=9\)

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

Is the graph of \(x^{2}+y^{2}=4\) the same as the graph of \(y^{2}+x^{2}=4\) ? Explain your answer.

Is the graph of \(x^{2}+y^{2}=0\) a circle? If so, what is the length of a radius?

Is the graph of \(4 x^{2}+9 y^{2}=36\) the same as the graph of \(9 x^{2}+4 y^{2}=36 ?\) Explain your answer.

\(y^{2}-8 y-x^{2}-4 x+3=0\)

\(4 x^{2}+24 x-y^{2}+4 y+28=0\)

\(x^{2}+y^{2}-2 x-6 y-6=0\)

\(x^{2}+y^{2}+4 x-12 y+39=0\)

\(x^{2}+y^{2}+6 x+10 y+18=0\)

\(x^{2}+y^{2}-10 x+2 y+1=0\)

\(x^{2}+y^{2}+4 x+14 y+50=0\)

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

The graphs of equations of the form \(x y=k\), where \(k\) is a nonzero constant, are also hyperbolas, sometimes referred to as rectangular hyperbolas. Graph each of the following. (a) \(x y=3\) (b) \(x y=5\) (c) \(x y=-2\) (d) \(x y=-4\)

\(x^{2}+y^{2}-16 x+6 y+71=0\)

What is the graph of \(x y=0\) ? Defend your answer.

We have graphed various equations of the form \(A x^{2}+\) \(B y^{2}=C\), where \(C\) is a nonzero constant. Now graph each of the following. (a) \(x^{2}+y^{2}=0\) (b) \(2 x^{2}+3 y^{2}=0\) (c) \(x^{2}-y^{2}=0\) (d) \(4 y^{2}-x^{2}=0\)

\(x^{2}+y^{2}+6 x-8 y=0\)

Explain the concept of an asymptote.

. \(x^{2}+y^{2}-16 x+30 y=0\)

How would you convince someone that \(y=(x+3)^{2}\) is the basic parabola moved 3 units to the left but that \(y=\) \((x-3)^{2}\) is the basic parabola moved 3 units to the right?

Explain how asymptotes can be used to help graph hyperbolas.

How does the graph of \(-y=x^{2}\) compare to the graph of \(y=x^{2}\) ? Explain your answer.

Are the graphs of \(x^{2}-y^{2}=0\) and \(y^{2}-x^{2}=0\) identical? Are the graphs of \(x^{2}-y^{2}=4\) and \(y^{2}-x^{2}=4\) identical? Explain your answers.

\(9 x^{2}+9 y^{2}-6 x-12 y-40=0\)

How does the graph of \(y=4 x^{2}\) compare to the graph of \(y=2 x^{2}\) ? Explain your answer.

(a) Graph \(y=x^{2}, y=2 x^{2}, y=3 x^{2}\), and \(y=4 x^{2}\) on the same set of axes. (b) Graph \(y=x^{2}, y=\frac{3}{4} x^{2}, y=\frac{1}{2} x^{2}\), and \(y=\frac{1}{5} x^{2}\) on the same set of axes. (c) Graph \(y=x^{2}, y=-x^{2}, y=-3 x^{2}\), and \(y=-\frac{1}{4} x^{2}\) on the same set of axes.

(a) Graph \(y=x^{2}, y=(x-2)^{2}, y=(x-3)^{2}\), and \(y=\) \((x-5)^{2}\) on the same set of axes. (b) Graph \(y=x^{2}, y=(x+1)^{2}, y=(x+3)^{2}\), and \(y=\) \((x+6)^{2}\) on the same set of axes.

For each of the following equations, (1) predict the type and location of the graph, and (2) use your graphing calculator to check your predictions. (a) \(x^{2}+y^{2}=100\) (b) \(x^{2}-y^{2}=100\) (c) \(y^{2}-x^{2}=100\) (d) \(y=-x^{2}+9\) (e) \(2 x^{2}+y^{2}=14\) (f) \(x^{2}+2 y^{2}=14\) (g) \(x^{2}+2 x+y^{2}-4=0\) (h) \(x^{2}+y^{2}-4 y-2=0\) (i) \(y=x^{2}+16\) (j) \(y^{2}=x^{2}+16\) (k) \(9 x^{2}-4 y^{2}=72\) (1) \(4 x^{2}-9 y^{2}=72\) (m) \(y^{2}=-x^{2}-4 x+6\)

(a) Graph \(y=x^{2}, y=(x-2)^{2}+3, y=(x+4)^{2}-2\), and \(y=(x-6)^{2}-4\) on the same set of axes. (b) Graph \(y=x^{2}, y=2(x+1)^{2}+4, y=3(x-1)^{2}-3\), and \(y=\frac{1}{2}(x-5)^{2}+2\) on the same set of axes. (c) Graph \(y=x^{2}, y=-(x-4)^{2}-3, y=-2(x+3)^{2}-1\), and \(y=-\frac{1}{2}(x-2)^{2}+6\) on the same set of axes.

\(x^{2}+y^{2}+6 x-2 y+6=0\)

(a) Graph \(y=x^{2}-12 x+41\) and \(y=x^{2}+12 x+41\) on the same set of axes. What relationship seems to exist between the two graphs? (b) Graph \(y=x^{2}-8 x+22\) and \(y=-x^{2}+8 x-22\) on the same set of axes. What relationship seems to exist between the two graphs? (c) Graph \(y=x^{2}+10 x+29\) and \(y=-x^{2}+10 x-29\) on the same set of axes. What relationship seems to exist between the two graphs? (d) Summarize your findings for parts (a) through (c).

\(x^{2}+y^{2}-4 x-6 y-12=0\)

\(x^{2}+y^{2}-4 x+3=0\)

\(x^{2}+y^{2}+4 x+4 y-8=0\)

\(x^{2}+y^{2}-6 x+6 y+2=0\)

Find the equation of the circle that passes through the origin and has its center at \((0,4)\).

Find the equation of the circle that passes through the origin and has its center at \((-6,0)\).

Find the equation of the circle that passes through the origin and has its center at \((-4,3)\).

Find the equation of the circle that passes through the origin and has its center at \((8,-15)\).

On which axis does the center of the circle \(x^{2}+y^{2}-\) \(8 y+7=0\) lie? Defend your answer.

Give a step-by-step description of how you would help someone graph the parabola \(y=2 x^{2}-12 x+9\).

The points \((x, y)\) and \((y, x)\) are mirror images of each other across the line \(y=x\). Therefore, by interchanging \(x\) and \(y\) in the equation \(y=a x^{2}+b x+c\), we obtain the equation of its mirror image across the line \(y=x\); namely, \(x=a y^{2}+b y+c\). Thus to graph \(x=y^{2}+2\), we can first graph \(y=x^{2}+2\) and then reflect it across the line \(y=x\), as indicated in Figure 8.22. Graph each of the following parabolas. (a) \(x=y^{2}\) (c) \(x=y^{2}-1\) (b) \(x=-y^{2}\) (d) \(x=-y^{2}+3\) (e) \(x=-2 y^{2}\) (f) \(x=3 y^{2}\) (g) \(x=y^{2}+4 y+7\) (h) \(x=y^{2}-2 y-3\)

By expanding \((x-h)^{2}+(y-k)^{2}=r^{2}\), we obtain \(x^{2}-\) \(2 h x+h^{2}+y^{2}-2 k y+k^{2}-r^{2}=0\). When we compare this result to the form \(x^{2}+y^{2}+D x+E y+F=0\), we see that \(D=-2 h, E=-2 k\), and \(F=h^{2}+k^{2}-r^{2}\). Therefore, the center and length of a radius of a circle can be found by using \(h=\frac{D}{-2}, k=\frac{E}{-2}\), and \(r=\sqrt{h^{2}+k^{2}-F}\). Use these relationships to find the center and the length of a radius of each of the following circles. (a) \(x^{2}+y^{2}-2 x-8 y+8=0\) (b) \(x^{2}+y^{2}+4 x-14 y+49=0\) (c) \(x^{2}+y^{2}+12 x+8 y-12=0\) (d) \(x^{2}+y^{2}-16 x+20 y+115=0\) (e) \(x^{2}+y^{2}-12 y-45=0\) (f) \(x^{2}+y^{2}+14 x=0\)

Graph each of the following parabolas and circles. Be sure to set your boundaries so that you get a complete graph. (a) \(x^{2}+24 x+y^{2}+135=0\) (b) \(y=x^{2}-4 x+18\) (c) \(x^{2}+y^{2}-18 y+56=0\) (d) \(x^{2}+y^{2}+24 x+28 y+336=0\) (e) \(y=-3 x^{2}-24 x-58\) (f) \(y=x^{2}-10 x+3\)