Chapter 7

Wre a graphing utility to graph \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0 .\) Is the graph a hyperbola? In general, what is the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 ?\)

Use a graphing utility to graph the parabolas in Exercises 77-78. Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+10 y-x+25=0$$

Write an equation for the path of each of the following elliptical orbits. Then use a graphing utility to graph the two cllipses in the same viewing rectangle. Can you see why early astronomers had difficulty detecting that these orbits are ellipses rather than circles? \(\cdot\) Earth's orbit: Length of major axis: 186 million miles Length of minor axis: 185.8 million miles \(\cdot\) Mars's orbit: Length of major axis: 283.5 million miles Length of minor axis: 278.5 million miles

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.

In Exercises 79-80, write each equation as a quadratic equation in \(y\) and then use the quadratic formula to express \(y\) in terms of \(x\). Graph the resulting two equations using a graphing utility. What effect does the \(xy\)-term have on the graph of the resulting parabola? $$16 x^{2}-24 x y+9 y^{2}-60 x-80 y+100=0$$

Write \(4 x^{2}-6 x y+2 y^{2}-3 x+10 y-6=0\) as a quadratic equation in \(y\) and then use the quadratic formula to express \(y\) in terms of \(x .\) Graph the resulting two equations using a graphing utility in a \([-50,70,10]\) by \([-30,50,10]\) viewing rectangle. What effect does the \(x y\) -term have on the graph of the resulting hyperbola? What problems would you encounter if you attempted to write the given equation in standard form by completing the square?

Graph \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\) and \(\frac{x|x|}{16}-\frac{y|y|}{9}=1\) in the same viewing rectangle. Explain why the graphs are not the same.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so I was able to graph the ellipse.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In a whispering gallery at our science museum, I stood at one focus, my friend stood at the other focus, and we had a clear conversation, very little of which was heard by the 25 museum visitors standing between us.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I noticed that depending on the values for \(A\) and \(B\), assuming that they are both not zero, the graph of \(A x^{2}+B y^{2}=C\) can represent any of the conic sections other than a parabola.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm using a telescope in which light from distant stars is reflected to the focus of a parabolic mirror.

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The parabola whose equation is \(x=2 y-y^{2}+5\) opens to the right.

What happens to the shape of the graph of \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) as \(\frac{c}{a} \rightarrow 0,\) where \(c^{2}=a^{2}-b^{2} ?\)

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. Some parabolas that open to the right have equations that define \(y\) as a function of \(x .\)

Will help you prepare for the material covered in the next section. Divide both sides of \(4 x^{2}-9 y^{2}=36\) by 36 and simplify. How does the simplified equation differ from that of an ellipse?

Will help you prepare for the material covered in the next section. Consider the equation \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\) a. Find the \(x\) -intercepts. b. Explain why there are no \(y\) -intercepts.

Will help you prepare for the material covered in the next section. Consider the equation \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\) a. Find the \(y\) -intercepts. b. Explain why there are no \(x\) -intercepts.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What happens to the shape of the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) as \(\frac{c}{a} \rightarrow \infty,\) where \(c^{2}=a^{2}+b^{2} ?\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find the standard form of the equation of the hyperbola with vertices \((5,-6)\) and \((5,6),\) passing through \((0,9)\)

Consult the research department of your library or the Internet to find an example of architecture that incorporates one or more conic sections in its design. Share this example with other group members. Explain precisely how conic sections are used. Do conic sections enhance the appeal of the architecture? In what ways?

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(\frac{(-1)^{n}}{3^{n}-1}\) for \(n=1,2,3,\) and 4

Graph each parabola with the given equation. \(y=x^{2}+4 x-5\)

Will help you prepare for the material covered in the first section of the next chapter. Find the product of all positive integers from \(n\) down through 1 for \(n=5\).

Graph each parabola with the given equation. \(y=-3(x-1)^{2}+2\)

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(i^{2}+1\) for all consecutive integers from 1 to 6 inclusive. Then find the sum of the six evaluations.

Isolate the terms involving \(y\) on the left side of the equation: $$ y^{2}+2 y+12 x-23-0 $$ Then write the equation in an equivalent form by completing the square on the left side.