Chapter 10

Identify the shape of the graph of the equation \(2 x^{2}+3 x-4 y+2=0\)

For Exercises \(40-43,\) use the following information. since a circle is not the graph of a function, you cannot enter its equation directly into a graphing calculator. Instead, you must solve the equation for \(y .\) The result will contain a \pm symbol, so you will have two functions. Solve \((x+3)^{2}+y^{2}=16\) for \(x .\) What parts of the circle do the two expressions for \(x\) represent?

Solve each equation. Round to the nearest ten-thousandth. $$ 3 e^{x}-2=0 $$

CHALLENGE For Exercises \(44-48,\) find all values of \(k\) for which the system of equations has the given number of solutions. If no values of \(k\) meet the condition, write none. $$ x^{2}+y^{2}=k^{2} \quad \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$ no solutions

Use the table below that shows the number of married Americans over the last few decades. $$\begin{array}{|c|c|c|c|c|c|}\hline 1980 & {1990} & {1995} & {1999} & {2000} & {2010} \\ \hline 104.6 & {112.6} & {116.7} & {118.9} & {120.2} & {?} \\\ \hline\end{array}$$ Predict the number of married Americans in 2010.

Write an equation for the ellipse that satisfies each set of conditions. endpoints of major axis at \((1,2)\) and \((9,2),\) endpoints of minor axis at \((5,1)\) and \((5,3)\)

What type of conic section is represented by the equation \(y^{2}-6 y=x^{2}-8 ?\)

CHALENGE The parabola with equation \(y=(x-4)^{2}+3\) has its vertex at \((4,3)\) and passes through \((5,4) .\) Find an equation of a different parabola with its vertex at \((4,3)\) and that passes through \((5,4) .\)

Solve each equation. Round to the nearest ten-thousandth. $$ e^{3 x}=4 $$

CHALLENGE For Exercises \(44-48,\) find all values of \(k\) for which the system of equations has the given number of solutions. If no values of \(k\) meet the condition, write none. $$ x^{2}+y^{2}=k^{2} \quad \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$ one solution

Graph the line with the given equation. \(y=2 x\)

Write an equation for the ellipse that satisfies each set of conditions. major axis 8 units long and parallel to \(y\) -axis, minor axis 6 units long, center at \((-3,1)\)

Write \(x^{2}+y^{2}+6 x-2 y-54=0\) in standard form by completing the square. Describe the transformation that can be applied to the graph of \(x^{2}+y^{2}=64\) to obtain the graph of the given equation.

Solve each equation. Round to the nearest ten-thousandth. $$ \ln (x+2)=5 $$

Graph the line with the given equation. \(y=-2 x\)

Write an equation for the ellipse that satisfies each set of conditions. foci at \((5,4)\) and \((-3,4),\) major axis 10 units long

REASONING Explain why the graph of the equation \(x^{2}+y^{2}-4 x+2 y+5=0\) is a single point.

Juwan says that the circle with equation \((x-4)^{2}+y^{2}=36\) has radius 36 units. Lucy says that the radius is 6 units. Who is correct? Explain your reasoning.

Write in the form \(y=a(x-h)^{2}+k\) $$ y=x^{2}+6 x+9 $$

Graph the line with the given equation. \(y=-\frac{1}{2} x\)

Find the center and radius of the circle with equation \(x^{2}+y^{2}-10 x+2 y+\) \(22=0 .\) Then graph the circle.

CHALLENGE For Exercises 47 and \(48,\) use the following information. The graph of an equation of the form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0\) is a special case of a hyperbola. Identify the graph of such an equation.

REVIEW \(\log _{9} 30=\) $$ \begin{array}{l}{\mathbf{F} \log _{10} 9+\log _{10} 30} \\ {\mathbf{G} \log _{10} 9-\log _{10} 30} \\ {\mathbf{H}\left(\log _{10} 9\right)\left(\log _{10} 30\right)} \\ {\mathbf{J} \frac{\log _{10} 30}{\log _{10} 9}}\end{array} $$

A circle has its center on the line with equation \(y=2 x .\) It passes through \((1,-3)\) and has a radius of \(\sqrt{5}\) units. Write an equation of the circle.

Write in the form \(y=a(x-h)^{2}+k\) $$ y=2 x^{2}+20 x+50 $$

CHALLENGE For Exercises \(44-48,\) find all values of \(k\) for which the system of equations has the given number of solutions. If no values of \(k\) meet the condition, write none. $$ x^{2}+y^{2}=k^{2} \quad \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$ four solutions

Graph the line with the given equation. \(y=\frac{1}{2} x\)

Solve each equation by factoring. $$ x^{2}+6 x+8=0 $$

Find the distance between each pair of points with the given coordinates. $$ (7,3),(-5,8) $$

Write in the form \(y=a(x-h)^{2}+k\) $$ y=-3 x^{2}-18 x-10 $$

Graph the line with the given equation. \(y+2=2(x-1)\)

Solve each equation by factoring. $$ 2 q^{2}+11 q=21 $$

Find the distance between each pair of points with the given coordinates. $$ (4,-1),(-2,7) $$

ACT/SAT What is the center of the circle with equation \(x^{2}+y^{2}-10 x+\) \(6 y+27=0 ?\) $$ \begin{array}{l}{\text { A }(-10,6)} \\ {\text { B }(1,1)} \\ {\text { C }(10,-6)} \\ {\text { D }(5,-3)}\end{array} $$

Graph the line with the given equation. \(y+2=-2(x-1)\)

Find the distance between each pair of points with the given coordinates. $$ (-3,1),(0,6) $$

REVIEW If the surface area of a cube is increased by a factor of \(9,\) how is the length of the side of the cube changed? \(\mathbf{F}\) It is 2 times the original length. \(\mathbf{G}\) It is 3 times the original length. \(\mathbf{H}\) It is 4 times the original length. \(\mathbf{J}\) It is 5 times the original length.

ACT/SAT How many solutions does the system of equations \(\frac{x^{2}}{5^{2}}-\frac{y^{2}}{3^{2}}=1\) and \((x-3)^{2}+y^{2}=9\) have? A 0 B 1 C 2 D 4

Solve \(|2 x+1|=9\)

RADIOACTVITY The decay of Radon-222 can be modeled by the equation \(y=a e^{-0.1313 t}\) where \(t\) is measured in days. What is the half-life of Radon- 222\(?\)

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ x=-3 y^{2}+1 $$

REVIEW Given: Two angles are supplementary. One angle is \(25^{\circ}\) more than the measure of the other angle. Conclusion: The measures of the angles are \(65^{\circ}\) and \(90^{\circ} .\) This conclusion \(-\) \(\mathrm{F}\) is contradicted by the first statement given. \(\mathrm{G}\) is verified by the first statement given. H invalidates itself because a \(90^{\circ}\) angle cannot be supplementary to another. J verifies itself because \(90^{\circ}\) is \(25^{\circ}\) more than \(65^{\circ} .\)

Simplify \(7 x+8 y+9 y-5 x\).

REVIEW The graph of \(\left(\frac{x}{4}\right)^{2}-\left(\frac{y}{5}\right)^{2}=1\) is a hyperbola. Which set of equations represents the asymptotes of the hyperbola's graph? $$ \begin{array}{l}{\mathbf{F} \quad y=\frac{4}{5} x, y=-\frac{4}{5} x} \\\ {\mathbf{G} y=\frac{1}{4} x, y=-\frac{1}{4} x} \\ {\mathbf{H} y=\frac{5}{4} x, y=-\frac{5}{4} x} \\ {\mathbf{J} \quad y=\frac{1}{5} x, y=-\frac{1}{5} x}\end{array} $$

HEATH Alisa's heart rate is usually 120 beats per minute when she runs. If she runs for 2 hours every day, about how many times will her heart beat during the amount of time she exercises in two weeks? Express in scientific notation.

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ y+2=-(x-3)^{2} $$

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+y^{2}+4 x+2 y-6=0 $$

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ 2 x^{2}+3 x y-5 y^{2}=0 $$

Write an equation of the hyperbola that satisfies each set of conditions. vertices \((5,10)\) and \((5,-2),\) conjugate axis of length 8 units

Simplify each radical expression. \(\sqrt{16}\)