Chapter 11 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

##

Worked example

circle

A circle is all points in a plane that are a fixed distance from a fixed point in the plane.

ellipse

An ellipse is all points in a plane where the sum of the distances from two fixed points is constant.

hyperbola

A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.

parabola

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.

system of nonlinear equations

A system of nonlinear equations is a system where at least one of the equations is not linear.

##

Key Concepts

• Distance Formula: The distance d between the two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is

• Midpoint Formula: The midpoint of the line segment whose endpoints are the two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is

To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.

• Circle: A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the center, $(h,k),$ and the fixed distance is called the radius, r, of the circle.

• Standard Form of the Equation a Circle: The standard form of the equation of a circle with center, $(h,k),$ and radius, r, is

• General Form of the Equation of a Circle: The general form of the equation of a circle is

• Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola. Vertical Parabolas General form$y=a{x}^{2}+bx+c$ Standard form$y=a{(x-h)}^{2}+k$Orientation $a>0$ up; $a<0$ down $a>0$ up; $a<0$ down Axis of symmetry $x=-\frac{b}{2a}$$x=h$ Vertex Substitute $x=-\frac{b}{2a}$ andsolve for y. $(h,k)$ y- intercept Let $x=0$ Let $x=0$ x-intercepts Let $y=0$ Let $y=0$

• How to graph vertical parabolas $(y=a{x}^{2}+bx+c$ or $f(x)=a{(x-h)}^{2}+k)$ using properties. Step 1. Determine whether the parabola opens upward or downward. Step 2. Find the axis of symmetry. Step 3. Find the vertex. Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. Step 5. Find the x-intercepts. Step 6. Graph the parabola. Horizontal Parabolas General form$x=a{y}^{2}+by+c$ Standard form$x=a{(y-k)}^{2}+h$Orientation $a>0$ right; $a<0$ left $a>0$ right; $a<0$ left Axis of symmetry $y=-\frac{b}{2a}$$y=k$ Vertex Substitute $y=-\frac{b}{2a}$ andsolve for x. $(h,k)$ y-intercepts Let $x=0$ Let $x=0$ x-intercept Let $y=0$ Let $y=0$

• How to graph horizontal parabolas $(x=a{y}^{2}+by+c$ or $x=a{(y-k)}^{2}+h)$ using properties. Step 1. Determine whether the parabola opens to the left or to the right. Step 2. Find the axis of symmetry. Step 3. Find the vertex. Step 4. Find the x-intercept. Find the point symmetric to the x-intercept across the axis of symmetry. Step 5. Find the y-intercepts. Step 6. Graph the parabola.

• Ellipse: An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse. If we draw a line through the foci intersects the ellipse in two points—each is called a vertex of the ellipse. The segment connecting the vertices is called the major axis. The midpoint of the segment is called the center of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.

• Standard Form of the Equation an Ellipse with Center $(0,0):$ The standard form of the equation of an ellipse with center $(0,0),$ is

The x-intercepts are $(a,0)$ and $(-a,0).$ The y-intercepts are $(0,b)$ and $(0,-b).$

• How to an Ellipse with Center $(0,0)$ Step 1. Write the equation in standard form. Step 2. Determine whether the major axis is horizontal or vertical. Step 3. Find the endpoints of the major axis. Step 4. Find the endpoints of the minor axis Step 5. Sketch the ellipse.

• Standard Form of the Equation an Ellipse with Center $(h,k):$ The standard form of the equation of an ellipse with center $(h,k),$ is

When $a>b,$ the major axis is horizontal so the distance from the center to the vertex is a. When $b>a,$ the major axis is vertical so the distance from the center to the vertex is b.

• Hyperbola: A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola. The line through the foci, is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola. Standard Forms of the Equation a Hyperbola with Center $(0,0)$$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$$\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$Orientation Transverse axis on the x-axis.Opens left and right Transverse axis on the y-axis.Opens up and down Vertices $(-a,0),$$(a,0)$$(0,-a),$$(0,a)$ x-intercepts $(-a,0),$$(a,0)$ none y-intercepts none $(0,-a)$, $(0,a)$ Rectangle Use $(\pm{}a,0)$$(0,\pm{}b)$ Use $(0,\pm{}a)$$(\pm{}b,0)$ asymptotes $y=\frac{b}{a}x,$$y=-\frac{b}{a}x$$y=\frac{a}{b}x,$$y=-\frac{a}{b}x$

• How to graph a hyperbola centered at $(0,0).$ Step 1. Write the equation in standard form. Step 2. Determine whether the transverse axis is horizontal or vertical. Step 3. Find the vertices. Step 4. Sketch the rectangle centered at the origin intersecting one axis at $\pm{}a$ and the other at $\pm{}b.$ Step 5. Sketch the asymptotes—the lines through the diagonals of the rectangle. Step 6. Draw the two branches of the hyperbola. Standard Forms of the Equation a Hyperbola with Center $(h,k)$$\frac{{(x-h)}^{2}}{{a}^{2}}-\frac{{(y-k)}^{2}}{{b}^{2}}=1$$\frac{{(y-k)}^{2}}{{a}^{2}}-\frac{{(x-h)}^{2}}{{b}^{2}}=1$Orientation Transverse axis is horizontal.Opens left and right Transverse axis is vertical.Opens up and down Center $(h,k)$$(h,k)$ Vertices a units to the left and right of the center a units above and below the center Rectangle Use a units left/right of center*b* units above/below the center Use a units above/below the center*b* units left/right of center

• How to graph a hyperbola centered at $(h,k).$ Step 1. Write the equation in standard form. Step 2. Determine whether the transverse axis is horizontal or vertical. Step 3. Find the center and $a,b.$ Step 4. Sketch the rectangle centered at $(h,k)$ using $a,b.$ Step 5. Sketch the asymptotes—the lines through the diagonals of the rectangle. Mark the vertices. Step 6. Draw the two branches of the hyperbola. Conic Characteristics of ${x}^{2}-$ and ${y}^{2}-$ terms ExampleParabola Either ${x}^{2}$ OR ${y}^{2}.$ Only one variable is squared. $x=3{y}^{2}-2y+1$ Circle ${x}^{2}-$ and ${y}^{2}-$ terms must have the same coefficients and they must be the same sign as the constant after the = sign ${x}^{2}+{y}^{2}=49$ Ellipse ${x}^{2}-$ and ${y}^{2}-$ terms have the same sign, different coefficients $4{x}^{2}+25{y}^{2}=100$ Hyperbola ${x}^{2}-$ and ${y}^{2}-$ terms have different signs $25{y}^{2}-4{x}^{2}=100$

• How to solve a system of nonlinear equations by graphing. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Graph the first equation. Step 3. Graph the second equation on the same rectangular coordinate system. Step 4. Determine whether the graphs intersect. Step 5. Identify the points of intersection. Step 6. Check that each ordered pair is a solution to both original equations.

• How to solve a system of nonlinear equations by substitution. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Solve one of the equations for either variable. Step 3. Substitute the expression from Step 2 into the other equation. Step 4. Solve the resulting equation. Step 5. Substitute each solution in Step 4 into one of the original equations to find the other variable. Step 6. Write each solution as an ordered pair. Step 7. Check that each ordered pair is a solution to both original equations.

• How to solve a system of equations by elimination. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Write both equations in standard form. Step 3. Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites. Step 4. Add the equations resulting from Step 3 to eliminate one variable. Step 5. Solve for the remaining variable. Step 6. Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable. Step 7. Write each solution as an ordered pair. Step 8. Check that each ordered pair is a solution to both original equations.

##

Review Exercises

Use the Distance Formula

In the following exercises, find the distance between the points. Round to the nearest tenth if needed.

Use the Midpoint Formula

In the following exercises, find the midpoint of the line segment whose endpoints are given.

Write the Equation of a Circle in Standard Form

In the following exercises, write the standard form of the equation of the circle with the given information.

Graph a Circle

In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.

Graph Vertical Parabolas

In the following exercises, graph each equation by using its properties.

In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.

Graph Horizontal Parabolas

In the following exercises, graph each equation by using its properties.

In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.

Solve Applications with Parabolas

In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

Graph an Ellipse with Center at the Origin

In the following exercises, graph each ellipse.

Find the Equation of an Ellipse with Center at the Origin

In the following exercises, find the equation of the ellipse shown in the graph.

Graph an Ellipse with Center Not at the Origin

In the following exercises, graph each ellipse.

In the following exercises, ⓐ write the equation in standard form and ⓑ graph.

Solve Applications with Ellipses

In the following exercises, write the equation of the ellipse described.

Graph a Hyperbola with Center at $(0,0)$

In the following exercises, graph.

Graph a Hyperbola with Center at $(h,k)$

In the following exercises, graph.

In the following exercises, ⓐ write the equation in standard form and ⓑ graph.

Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola

In the following exercises, identify the type of graph.

Solve a System of Nonlinear Equations Using Graphing

In the following exercises, solve the system of equations by using graphing.

Solve a System of Nonlinear Equations Using Substitution

In the following exercises, solve the system of equations by using substitution.

Solve a System of Nonlinear Equations Using Elimination

In the following exercises, solve the system of equations by using elimination.

Use a System of Nonlinear Equations to Solve Applications

In the following exercises, solve the problem using a system of equations.

##

In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.

In the following exercises, write the standard form of the equation of the circle with the given information.

In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and ⓑ graph the equation.

In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation.

Key Terms

circle

A circle is all points in a plane that are a fixed distance from a fixed point in the plane.

ellipse

An ellipse is all points in a plane where the sum of the distances from two fixed points is constant.

hyperbola

A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.

parabola

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.

system of nonlinear equations

A system of nonlinear equations is a system where at least one of the equations is not linear.

##

Key Concepts

• Distance Formula: The distance d between the two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is

• Midpoint Formula: The midpoint of the line segment whose endpoints are the two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is

To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.

• Circle: A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the center, $(h,k),$ and the fixed distance is called the radius, r, of the circle.

• Standard Form of the Equation a Circle: The standard form of the equation of a circle with center, $(h,k),$ and radius, r, is

• General Form of the Equation of a Circle: The general form of the equation of a circle is

• Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola. Vertical Parabolas General form$y=a{x}^{2}+bx+c$ Standard form$y=a{(x-h)}^{2}+k$Orientation $a>0$ up; $a<0$ down $a>0$ up; $a<0$ down Axis of symmetry $x=-\frac{b}{2a}$$x=h$ Vertex Substitute $x=-\frac{b}{2a}$ andsolve for y. $(h,k)$ y- intercept Let $x=0$ Let $x=0$ x-intercepts Let $y=0$ Let $y=0$

• How to graph vertical parabolas $(y=a{x}^{2}+bx+c$ or $f(x)=a{(x-h)}^{2}+k)$ using properties. Step 1. Determine whether the parabola opens upward or downward. Step 2. Find the axis of symmetry. Step 3. Find the vertex. Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. Step 5. Find the x-intercepts. Step 6. Graph the parabola. Horizontal Parabolas General form$x=a{y}^{2}+by+c$ Standard form$x=a{(y-k)}^{2}+h$Orientation $a>0$ right; $a<0$ left $a>0$ right; $a<0$ left Axis of symmetry $y=-\frac{b}{2a}$$y=k$ Vertex Substitute $y=-\frac{b}{2a}$ andsolve for x. $(h,k)$ y-intercepts Let $x=0$ Let $x=0$ x-intercept Let $y=0$ Let $y=0$

• How to graph horizontal parabolas $(x=a{y}^{2}+by+c$ or $x=a{(y-k)}^{2}+h)$ using properties. Step 1. Determine whether the parabola opens to the left or to the right. Step 2. Find the axis of symmetry. Step 3. Find the vertex. Step 4. Find the x-intercept. Find the point symmetric to the x-intercept across the axis of symmetry. Step 5. Find the y-intercepts. Step 6. Graph the parabola.

• Ellipse: An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse. If we draw a line through the foci intersects the ellipse in two points—each is called a vertex of the ellipse. The segment connecting the vertices is called the major axis. The midpoint of the segment is called the center of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.

• Standard Form of the Equation an Ellipse with Center $(0,0):$ The standard form of the equation of an ellipse with center $(0,0),$ is

The x-intercepts are $(a,0)$ and $(-a,0).$ The y-intercepts are $(0,b)$ and $(0,-b).$

• How to an Ellipse with Center $(0,0)$ Step 1. Write the equation in standard form. Step 2. Determine whether the major axis is horizontal or vertical. Step 3. Find the endpoints of the major axis. Step 4. Find the endpoints of the minor axis Step 5. Sketch the ellipse.

• Standard Form of the Equation an Ellipse with Center $(h,k):$ The standard form of the equation of an ellipse with center $(h,k),$ is

When $a>b,$ the major axis is horizontal so the distance from the center to the vertex is a. When $b>a,$ the major axis is vertical so the distance from the center to the vertex is b.

• Hyperbola: A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola. The line through the foci, is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola. Standard Forms of the Equation a Hyperbola with Center $(0,0)$$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$$\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$Orientation Transverse axis on the x-axis.Opens left and right Transverse axis on the y-axis.Opens up and down Vertices $(-a,0),$$(a,0)$$(0,-a),$$(0,a)$ x-intercepts $(-a,0),$$(a,0)$ none y-intercepts none $(0,-a)$, $(0,a)$ Rectangle Use $(\pm{}a,0)$$(0,\pm{}b)$ Use $(0,\pm{}a)$$(\pm{}b,0)$ asymptotes $y=\frac{b}{a}x,$$y=-\frac{b}{a}x$$y=\frac{a}{b}x,$$y=-\frac{a}{b}x$

• How to graph a hyperbola centered at $(0,0).$ Step 1. Write the equation in standard form. Step 2. Determine whether the transverse axis is horizontal or vertical. Step 3. Find the vertices. Step 4. Sketch the rectangle centered at the origin intersecting one axis at $\pm{}a$ and the other at $\pm{}b.$ Step 5. Sketch the asymptotes—the lines through the diagonals of the rectangle. Step 6. Draw the two branches of the hyperbola. Standard Forms of the Equation a Hyperbola with Center $(h,k)$$\frac{{(x-h)}^{2}}{{a}^{2}}-\frac{{(y-k)}^{2}}{{b}^{2}}=1$$\frac{{(y-k)}^{2}}{{a}^{2}}-\frac{{(x-h)}^{2}}{{b}^{2}}=1$Orientation Transverse axis is horizontal.Opens left and right Transverse axis is vertical.Opens up and down Center $(h,k)$$(h,k)$ Vertices a units to the left and right of the center a units above and below the center Rectangle Use a units left/right of center*b* units above/below the center Use a units above/below the center*b* units left/right of center

• How to graph a hyperbola centered at $(h,k).$ Step 1. Write the equation in standard form. Step 2. Determine whether the transverse axis is horizontal or vertical. Step 3. Find the center and $a,b.$ Step 4. Sketch the rectangle centered at $(h,k)$ using $a,b.$ Step 5. Sketch the asymptotes—the lines through the diagonals of the rectangle. Mark the vertices. Step 6. Draw the two branches of the hyperbola. Conic Characteristics of ${x}^{2}-$ and ${y}^{2}-$ terms ExampleParabola Either ${x}^{2}$ OR ${y}^{2}.$ Only one variable is squared. $x=3{y}^{2}-2y+1$ Circle ${x}^{2}-$ and ${y}^{2}-$ terms must have the same coefficients and they must be the same sign as the constant after the = sign ${x}^{2}+{y}^{2}=49$ Ellipse ${x}^{2}-$ and ${y}^{2}-$ terms have the same sign, different coefficients $4{x}^{2}+25{y}^{2}=100$ Hyperbola ${x}^{2}-$ and ${y}^{2}-$ terms have different signs $25{y}^{2}-4{x}^{2}=100$

• How to solve a system of nonlinear equations by graphing. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Graph the first equation. Step 3. Graph the second equation on the same rectangular coordinate system. Step 4. Determine whether the graphs intersect. Step 5. Identify the points of intersection. Step 6. Check that each ordered pair is a solution to both original equations.

• How to solve a system of nonlinear equations by substitution. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Solve one of the equations for either variable. Step 3. Substitute the expression from Step 2 into the other equation. Step 4. Solve the resulting equation. Step 5. Substitute each solution in Step 4 into one of the original equations to find the other variable. Step 6. Write each solution as an ordered pair. Step 7. Check that each ordered pair is a solution to both original equations.

• How to solve a system of equations by elimination. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Write both equations in standard form. Step 3. Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites. Step 4. Add the equations resulting from Step 3 to eliminate one variable. Step 5. Solve for the remaining variable. Step 6. Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable. Step 7. Write each solution as an ordered pair. Step 8. Check that each ordered pair is a solution to both original equations.

##

Review Exercises

Use the Distance Formula

In the following exercises, find the distance between the points. Round to the nearest tenth if needed.

Use the Midpoint Formula

In the following exercises, find the midpoint of the line segment whose endpoints are given.

Write the Equation of a Circle in Standard Form

In the following exercises, write the standard form of the equation of the circle with the given information.

Graph a Circle

In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.

Graph Vertical Parabolas

In the following exercises, graph each equation by using its properties.

In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.

Graph Horizontal Parabolas

In the following exercises, graph each equation by using its properties.

In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.

Solve Applications with Parabolas

In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

Graph an Ellipse with Center at the Origin

In the following exercises, graph each ellipse.

Find the Equation of an Ellipse with Center at the Origin

In the following exercises, find the equation of the ellipse shown in the graph.

Graph an Ellipse with Center Not at the Origin

In the following exercises, graph each ellipse.

In the following exercises, ⓐ write the equation in standard form and ⓑ graph.

Solve Applications with Ellipses

In the following exercises, write the equation of the ellipse described.

Graph a Hyperbola with Center at $(0,0)$

In the following exercises, graph.

Graph a Hyperbola with Center at $(h,k)$

In the following exercises, graph.

In the following exercises, ⓐ write the equation in standard form and ⓑ graph.

Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola

In the following exercises, identify the type of graph.

Solve a System of Nonlinear Equations Using Graphing

In the following exercises, solve the system of equations by using graphing.

Solve a System of Nonlinear Equations Using Substitution

In the following exercises, solve the system of equations by using substitution.

Solve a System of Nonlinear Equations Using Elimination

In the following exercises, solve the system of equations by using elimination.

Use a System of Nonlinear Equations to Solve Applications

In the following exercises, solve the problem using a system of equations.

##

In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.

In the following exercises, write the standard form of the equation of the circle with the given information.

In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and ⓑ graph the equation.

In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation.

1. $(-2,5)$ and $(1,5)$

2. $(1,-4)$ and $(5,-5)$

3. $(3,7)$ and $(5,1)$

4. $(-3,2)$ and $(6,-9)$

1. radius is $\sqrt{7}$ and center is $(0,0)$

2. radius is 7 and center is $(-2,-5)$

3. center is $(2,2)$ and a point on the circle is $(4,4)$

4. $3{x}^{2}+3{y}^{2}=432$

1. ${(x+2)}^{2}+{(y+5)}^{2}=49$

2. ${x}^{2}+{y}^{2}-4y-60=0$

3. $y=2{x}^{2}+10x+7$

4. $y=-{x}^{2}+10x$

1. $y=2{x}^{2}-4x-2$

2. $y=-{x}^{2}+12x-35$

3. $x=2{y}^{2}+4y+6$

4. $x=-3{y}^{2}$

1. $x={y}^{2}+4y+5$

2. $x=-2{y}^{2}+4y$

3. $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{81}=1$

4. $9{x}^{2}+{y}^{2}=9$

1. $\frac{{(x+4)}^{2}}{16}+\frac{{(y+1)}^{2}}{9}=1$

2. $\frac{{(x+3)}^{2}}{9}+\frac{{(y-2)}^{2}}{25}=1$

3. $25{x}^{2}+4{y}^{2}-150x-56y+321=0$

4. $4{x}^{2}+9{y}^{2}-126y+405=0$

1. $\frac{{x}^{2}}{25}-\frac{{y}^{2}}{9}=1$

2. $9{y}^{2}-16{x}^{2}=144$

3. $\frac{{(x+1)}^{2}}{4}-\frac{{(y+1)}^{2}}{9}=1$

4. $\frac{{(y+2)}^{2}}{9}-\frac{{(x+1)}^{2}}{9}=1$

1. $4{x}^{2}-16{y}^{2}+8x+96y-204=0$

2. $4{y}^{2}-16{x}^{2}+32x-8y-76=0$

3. ⓐ $16{y}^{2}-9{x}^{2}-36x-96y-36=0$ⓑ ${x}^{2}+{y}^{2}-4x+10y-7=0$ⓒ $y={x}^{2}-2x+3$ⓓ $25{x}^{2}+9{y}^{2}=225$

4. $\mathrm{\{}\begin{array}{l} 3{x}^{2}-y=0 \\ y=2x-1 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=169 \\ x=12 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} y={x}^{2}+3 \\ y=-2x+2 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 9{x}^{2}+4{y}^{2}=36 \\ y-x=5 \\ \end{array}$

4. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=16 \\ {x}^{2}-2y-1=0 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 4{x}^{2}+9{y}^{2}=36 \\ 3{y}^{2}-4x=12 \\ \end{array}$

2. The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.

3. The perimeter of a rectangle is 58 meters and its area is 210 square meters. Find the length and width of the rectangle.

4. $(-4,-3)$ and $(-10,-11)$

1. radius is 11 and center is $(0,0)$

2. center is $(-2,3)$ and a point on the circle is $(2,-3)$

3. $4{x}^{2}+49{y}^{2}=196$

4. $3{x}^{2}+3{y}^{2}=27$

1. $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{81}=1$

2. $64{x}^{2}-9{y}^{2}=576$

3. ${x}^{2}+{y}^{2}+10x+6y+30=0$

4. $9{x}^{2}-25{y}^{2}-36x-50y-214=0$

1. Solve the nonlinear system of equations by graphing:$\mathrm{\{}\begin{array}{l} 3{y}^{2}-x=0 \\ y=-2x-1 \\ \end{array}.$

2. Solve the nonlinear system of equations using elimination:$\mathrm{\{}\begin{array}{l} {x}^{2}+9{y}^{2}=9 \\ 2{x}^{2}-9{y}^{2}=18 \\ \end{array}.$

3. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 20 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

4. For her birthday, Olive’s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is 55”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1452 square inches. Her entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Olive’s entertainment center?