91Ó°ÊÓ

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed. For an ellipse, the relationship between \(a, b,\) and \(c\) is given by the foci equation______since \(c

(a) graph the curves defined by the parametric equations using the specified interval and identify the graph (if possible) and (b) eliminate the parameter (Exercises \(7 \text { to } 16 \text { only })\) and write the corresponding rectangular form. $$\begin{aligned}&x=t-3 ; t \in[-5,5]\\\&y=2-0.5 t^{2}\end{aligned}$$

(a) graph the curves defined by the parametric equations using the specified interval and identify the graph (if possible) and (b) eliminate the parameter (Exercises \(7 \text { to } 16 \text { only })\) and write the corresponding rectangular form. $$\begin{aligned}&x=\frac{t^{3}}{10} ; t \in[-5,5]\\\&y=|t|\end{aligned}$$

Identify the center and radius of each circle, then sketch its graph. $$x^{2}+y^{2}+4 x+6 y-3=0$$

Sketch the graph of each ellipse. $$\frac{(x-2)^{2}}{25}+\frac{(y+3)^{2}}{4}=1$$

The Perpendicular Distance from a Point to a Line: \(d=\left|\frac{A x_{1}+B y_{1}+C}{\sqrt{A^{2}+B^{2}}}\right|\) The perpendicular distance from a point \(\left(x_{1}, y_{1}\right)\) to a given line can be found using the formula shown, where \(A x+B y+C=0\) is the equation of the line in standard form \((A, B, \text { and } C\) are integers). Use the formula to verify that \(P(-6,2)\) and \(Q(6,4)\) are an equal distance from the line \(y=-\frac{1}{2} x+3\) .

Complete the square in both \(x\) and \(y\) to write each equation in standard form. Then draw a complete graph of the relation and identify all important features. $$x^{2}+4 y^{2}-8 y+4 x-8=0$$

The Folium of Descartes: \(x(t)=\frac{3 k t}{1+t^{3}} ; y(t)=\frac{3 k t^{2}}{1+t^{3}}\) The Folium of Descartes is a parametric curve developed by Descartes in order to test the ability of Fermat to find its maximum and minimum values. a. Graph the curve on a graphing calculator with \(k=1\) using a reduced window \((\text { zoom } 4),\) with Tmin \(=-6,\) Tmax \(=6,\) and Tstep \(=0.1\) Locate the coordinates of the tip of the folium (the loop). b. This graph actually has a discontinuity (a break in the graph). At what value of \(t\) does this occur? c. Experiment with different values of \(k\) and generalize its effect on the basic graph.

The Witch of Agnesi: \(x(t)=2 k t ; y(t)=\frac{2 k}{1+t^{2}}\)The Witch of Agnesi is a parametric curve named by Maria Agnesi in \(1748 .\) Some believe she confused the Italian word for witch (versiera), with a similar word that meant free to move. In any case, the name stuck. The curve can also be stated in trigonometric form: \(x(t)=2 k \cot t\) and \(y=2 k \sin ^{2} t\) . a. Graph the curve with \(k=1\) on a calculator or computer on a reduced window ( \((200 \mathrm{m})\) 4) using both of the forms shown with Tmin \(=-6,\) Tmax \(=6,\) and Tstep \(=0.1 .\) Try to determine the maximum value. b. Explain why the \(x\) -axis is a horizontal asymptote. c. Experiment with different values of \(k\) and generalize its effect on the basic graph.

Find the vertex, focus, and directrix for the parabolas defined by the equations given, then use this information to sketch a complete graph (illustrate and name these features). For Exercises 43 to 60 , also include the focal chord. $$y^{2}=20 x$$