22.7: Review Exercises

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Chapter Review Exercises

Distance and Midpoint Formulas; Circles

Exercise (PageIndex{1}) Use the Distance Formula

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

1. ((-5,1)) and ((-1,4))
2. ((-2,5)) and ((1,5))
3. ((8,2)) and ((-7,-3))
4. ((1,-4)) and ((5,-5))

2. (d=3)

4. (d=sqrt{17}, d approx 4.1)

Exercise (PageIndex{2}) Use the Midpoint Formula

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

1. ((-2,-6)) and ((-4,-2))
2. ((3,7)) and ((5,1))
3. ((-8,-10)) and ((9,5))
4. ((-3,2)) and ((6,-9))

2. ((4,4))

4. (left(frac{3}{2},-frac{7}{2} ight))

Exercise (PageIndex{3}) 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.

1. radius is (15) and center is ((0,0))
2. radius is (sqrt{7}) and center is ((0,0))
3. radius is (9) and center is ((-3,5))
4. radius is (7) and center is ((-2,-5))
5. center is ((3,6)) and a point on the circle is ((3,-2))
6. center is ((2,2)) and a point on the circle is ((4,4))

2. (x^{2}+y^{2}=7)

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

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

Exercise (PageIndex{4}) Graph a Circle

In the following exercises,

1. Find the center and radius, then
2. Graph each circle.
1. (2 x^{2}+2 y^{2}=450)
2. (3 x^{2}+3 y^{2}=432)
3. ((x+3)^{2}+(y-5)^{2}=81)
4. ((x+2)^{2}+(y+5)^{2}=49)
5. (x^{2}+y^{2}-6 x-12 y-19=0)
6. (x^{2}+y^{2}-4 y-60=0)

2.

1. radius: (12,) center: ((0,0))

4.

1. radius: (7,) center: ((-2,-5))

6.

1. radius: (8,) center: ((0,2))

Parabolas

Exercise (PageIndex{5}) Graph Vertical Parabolas

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

1. (y=x^{2}+4 x-3)
2. (y=2 x^{2}+10 x+7)
3. (y=-6 x^{2}+12 x-1)
4. (y=-x^{2}+10 x)

2.

4.

Exercise (PageIndex{6}) Graph Vertical Parabolas

In the following exercises,

1. Write the equation in standard form, then
2. Use properties of the standard form to graph the equation.
1. (y=x^{2}+4 x+7)
2. (y=2 x^{2}-4 x-2)
3. (y=-3 x^{2}-18 x-29)
4. (y=-x^{2}+12 x-35)

2.

1. (y=2(x-1)^{2}-4)

4.

1. (y=-(x-6)^{2}+1)

Exercise (PageIndex{7}) Graph Horizontal Parabolas

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

1. (x=2 y^{2})
2. (x=2 y^{2}+4 y+6)
3. (x=-y^{2}+2 y-4)
4. (x=-3 y^{2})

2.

4.

Exercise (PageIndex{8}) Graph Horizontal Parabolas

In the following exercises,

1. Write the equation in standard form, then
2. Use properties of the standard form to graph the equation.
1. (x=4 y^{2}+8 y)
2. (x=y^{2}+4 y+5)
3. (x=-y^{2}-6 y-7)
4. (x=-2 y^{2}+4 y)

2.

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

4.

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

Exercise (PageIndex{9}) 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.

1.

2.

2. (y=-frac{1}{9} x^{2}+frac{10}{3} x)

Ellipses

Exercise (PageIndex{10}) Graph an Ellipse with Center at the Origin

In the following exercises, graph each ellipse.

1. (frac{x^{2}}{36}+frac{y^{2}}{25}=1)
2. (frac{x^{2}}{4}+frac{y^{2}}{81}=1)
3. (49 x^{2}+64 y^{2}=3136)
4. (9 x^{2}+y^{2}=9)

2.

4.

Exercise (PageIndex{11}) 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.

1.

2.

2. (frac{x^{2}}{36}+frac{y^{2}}{64}=1)

Exercise (PageIndex{12}) Graph an Ellipse with Center Not at the Origin

In the following exercises, graph each ellipse.

1. (frac{(x-1)^{2}}{25}+frac{(y-6)^{2}}{4}=1)
2. (frac{(x+4)^{2}}{16}+frac{(y+1)^{2}}{9}=1)
3. (frac{(x-5)^{2}}{16}+frac{(y+3)^{2}}{36}=1)
4. (frac{(x+3)^{2}}{9}+frac{(y-2)^{2}}{25}=1)

2.

4.

Exercise (PageIndex{13}) Graph an Ellipse with Center Not at the Origin

In the following exercises,

1. Write the equation in standard form and
2. Graph.
1. (x^{2}+y^{2}+12 x+40 y+120=0)
2. (25 x^{2}+4 y^{2}-150 x-56 y+321=0)
3. (25 x^{2}+4 y^{2}+150 x+125=0)
4. (4 x^{2}+9 y^{2}-126 x+405=0)

2.

1. (frac{(x-3)^{2}}{4}+frac{(y-7)^{2}}{25}=1)

4.

1. (frac{x^{2}}{9}+frac{(y-7)^{2}}{4}=1)

Exercise (PageIndex{14}) Solve Applications with Ellipses

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

1. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately (10) AU and the furthest is approximately (90) 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.

1. Solve

Hyperbolas

Exercise (PageIndex{15}) Graph a Hyperbola with Center at ((0,0))

In the following exercises, graph.

1. (frac{x^{2}}{25}-frac{y^{2}}{9}=1)
2. (frac{y^{2}}{49}-frac{x^{2}}{16}=1)
3. (9 y^{2}-16 x^{2}=144)
4. (16 x^{2}-4 y^{2}=64)

1.

3.

Exercise (PageIndex{16}) Graph a Hyperbola with Center at ((h,k))

In the following exercises, graph.

1. (frac{(x+1)^{2}}{4}-frac{(y+1)^{2}}{9}=1)
2. (frac{(x-2)^{2}}{4}-frac{(y-3)^{2}}{16}=1)
3. (frac{(y+2)^{2}}{9}-frac{(x+1)^{2}}{9}=1)
4. (frac{(y-1)^{2}}{25}-frac{(x-2)^{2}}{9}=1)

1.

3.

Exercise (PageIndex{17}) Graph a Hyperbola with Center at ((h,k))

In the following exercises,

1. Write the equation in standard form and
2. Graph.
1. (4 x^{2}-16 y^{2}+8 x+96 y-204=0)
2. (16 x^{2}-4 y^{2}-64 x-24 y-36=0)
3. (4 y^{2}-16 x^{2}+32 x-8 y-76=0)
4. (36 y^{2}-16 x^{2}-96 x+216 y-396=0)

1.

1. (frac{(x+1)^{2}}{16}-frac{(y-3)^{2}}{4}=1)

3.

1. (frac{(y-1)^{2}}{16}-frac{(x-1)^{2}}{4}=1)

Exercise (PageIndex{18}) Identify the Graph of Each Equation as a Circle, Parabola, Ellipse, or Hyperbola

In the following exercises, identify the type of graph.

1. (16 y^{2}-9 x^{2}-36 x-96 y-36=0)
2. (x^{2}+y^{2}-4 x+10 y-7=0)
3. (y=x^{2}-2 x+3)
4. (25 x^{2}+9 y^{2}=225)
1. (x^{2}+y^{2}+4 x-10 y+25=0)
2. (y^{2}-x^{2}-4 y+2 x-6=0)
3. (x=-y^{2}-2 y+3)
4. (16 x^{2}+9 y^{2}=144)

1.

1. Hyperbola
2. Circle
3. Parabola
4. Ellipse

Solve Systems of Nonlinear Equations

Exercise (PageIndex{19}) Solve a System of Nonlinear Equations Using Graphing

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

1. (left{egin{array}{l}{3 x^{2}-y=0} {y=2 x-1}end{array} ight.)
2. (left{egin{array}{l}{y=x^{2}-4} {y=x-4}end{array} ight.)
3. (left{egin{array}{l}{x^{2}+y^{2}=169} {x=12}end{array} ight.)
4. (left{egin{array}{l}{x^{2}+y^{2}=25} {y=-5}end{array} ight.)

1.

3.

Exercise (PageIndex{20}) Solve a System of Nonlinear Equations Using Substitution

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

1. (left{egin{array}{l}{y=x^{2}+3} {y=-2 x+2}end{array} ight.)
2. (left{egin{array}{l}{x^{2}+y^{2}=4} {x-y=4}end{array} ight.)
3. (left{egin{array}{l}{9 x^{2}+4 y^{2}=36} {y-x=5}end{array} ight.)
4. (left{egin{array}{l}{x^{2}+4 y^{2}=4} {2 x-y=1}end{array} ight.)

1. ((-1,4))

3. No solution

Exercise (PageIndex{21}) Solve a System of Nonlinear Equations Using Elimination

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

1. (left{egin{array}{l}{x^{2}+y^{2}=16} {x^{2}-2 y-1=0}end{array} ight.)
2. (left{egin{array}{l}{x^{2}-y^{2}=5} {-2 x^{2}-3 y^{2}=-30}end{array} ight.)
3. (left{egin{array}{l}{4 x^{2}+9 y^{2}=36} {3 y^{2}-4 x=12}end{array} ight.)
4. (left{egin{array}{l}{x^{2}+y^{2}=14} {x^{2}-y^{2}=16}end{array} ight.)

1. ((-sqrt{7}, 3),(sqrt{7}, 3))

3. ((-3,0),(0,-2),(0,2))

Exercise (PageIndex{22}) Use a System of Nonlinear Equations to Solve Applications

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

1. The sum of the squares of two numbers is (25). The difference of the numbers is (1). Find the numbers.
2. The difference of the squares of two numbers is (45). The difference of the square of the first number and twice the square of the second number is (9). 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. Colton purchased a larger microwave for his kitchen. The diagonal of the front of the microwave measures (34) inches. The front also has an area of (480) square inches. What are the length and width of the microwave?

1. (-3) and (-4) or (4) and (3)

3. If the length is (14) inches, the width is (15) inches. If the length is (15) inches, the width is (14) inches.

Practice Test

Exercise (PageIndex{23})

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.

1. ((-4,-3)) and ((-10,-11))
2. ((6,8)) and ((-5,-3))

1. distance: (10,) midpoint: ((-7,-7))

Exercise (PageIndex{24})

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

1. radius is (11) and center is ((0,0))
2. radius is (12) and center is ((10,-2))
3. center is ((-2,3)) and a point on the circle is ((2,-3))
4. Find the equation of the ellipse shown in the graph.

1. (x^{2}+y^{2}=121)

3. ((x+2)^{2}+(y-3)^{2}=52)

Exercise (PageIndex{25})

In the following exercises,

1. Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and
2. Graph the equation.
1. (4 x^{2}+49 y^{2}=196)
2. (y=3(x-2)^{2}-2)
3. (3 x^{2}+3 y^{2}=27)
4. (frac{y^{2}}{100}-frac{x^{2}}{36}=1)
5. (frac{x^{2}}{16}+frac{y^{2}}{81}=1)
6. (x=2 y^{2}+10 y+7)
7. (64 x^{2}-9 y^{2}=576)

1.

1. Ellipse

3.

1. Circle

5.

1. Ellipse

7.

1. Hyperbola

Exercise (PageIndex{26})

In the following exercises,

1. Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola,
2. Write the equation in standard form, and
3. Graph the equation.
1. (25 x^{2}+64 y^{2}+200 x-256 y-944=0)
2. (x^{2}+y^{2}+10 x+6 y+30=0)
3. (x=-y^{2}+2 y-4)
4. (9 x^{2}-25 y^{2}-36 x-50 y-214=0)
5. (y=x^{2}+6 x+8)
6. Solve the nonlinear system of equations by graphing: (left{egin{array}{l}{3 y^{2}-x=0} {y=-2 x-1}end{array} ight.).
7. Solve the nonlinear system of equations using substitution: (left{egin{array}{l}{x^{2}+y^{2}=8} {y=-x-4}end{array} ight.).
8. Solve the nonlinear system of equations using elimination: (left{egin{array}{l}{x^{2}+9 y^{2}=9} {2 x^{2}-9 y^{2}=18}end{array} ight.)
9. Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in (y=a x^{2}+b x+c) form.

10. 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. Use the graph to write an equation for the elliptical orbit of the comet.

11. The sum of two numbers is (22) and the product is (−240). Find the numbers.

12. 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?

2.

1. Circle
2. ((x+5)^{2}+(y+3)^{2}=4)

4.

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

6. No solution

8. ((0,-3),(0,3))

10. (frac{x^{2}}{2025}+frac{y^{2}}{1400}=1)

12. The length is (44) inches and the width is (33) inches. The TV will fit Olive’s entertainment center.

Glossary

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

22.7: Review Exercises

Find the first 5 terms of the sequence as well as the 30 th term.

Find the first 5 terms of the sequence.

a n = a n − 1 + 5 where a 1 = 0

a n = 4 a n − 1 + 1 where a 1 = − 2

a n = a n − 2 − 3 a n − 1 where a 1 = 0 and a 2 = − 3

a n = 5 a n − 2 − a n − 1 where a 1 = − 1 and a 2 = 0

Find the indicated partial sum.

Arithmetic Sequences and Series

Write the first 5 terms of the arithmetic sequence given its first term and common difference. Find a formula for its general term.

Given the terms of an arithmetic sequence, find a formula for the general term.

Calculate the indicated sum given the formula for the general term of an arithmetic sequence.

Find the sum of the first 175 positive odd integers.

Find the sum of the first 175 positive even integers.

Find all arithmetic means between a 1 = 2 3 and a 5 = − 2 3

Find all arithmetic means between a 3 = − 7 and a 7 = 13 .

A 5-year salary contract offers $58,200 for the first year with a$4,200 increase each additional year. Determine the total salary obligation over the 5-year period.

The first row of seating in a theater consists of 10 seats. Each successive row consists of four more seats than the previous row. If there are 14 rows, how many total seats are there in the theater?

Geometric Sequences and Series

Write the first 5 terms of the geometric sequence given its first term and common ratio. Find a formula for its general term.

Given the terms of a geometric sequence, find a formula for the general term.

a 2 = − 5 2 and a 5 = − 625 16

Find all geometric means between a 1 = − 1 and a 4 = 64 .

Find all geometric means between a 3 = 6 and a 6 = 162 .

Calculate the indicated sum given the formula for the general term of a geometric sequence.

Background

Bipolar disorder is a chronic condition characterized by elevated (manic) and depressive episodes often associated with difficulty functioning and poor quality of life. A diagnosis of bipolar disorder is also associated with an increased risk of cardiovascular disease leading to premature mortality (Roshanaei-Moghaddam and Katon, 2009 Dome et al., 2012 Crump et al., 2013). Further, obesity and a sedentary lifestyle are risk factors for diabetes, metabolic syndrome and cardiovascular disease, all of which disproportionally affect people with bipolar disorder (Elmslie et al., 2001 Morriss and Mohammed, 2005 Alsuwaidan et al., 2009 Cairney et al., 2009 Kilbourne et al., 2009). Thus, individuals with bipolar disorder face the dual struggle of needing to focus their attention and treatment on not only their mental health but also their physical health.

Exercise may be an excellent candidate to meet this need. Exercise unequivocally improves physical heath (e.g., obesity, cardiorespiratory fitness, blood pressure, cholesterol Cornelissen and Fagard, 2005 Church et al., 2007 Department of Health, 2011), but recent data also suggest that exercise is an effective treatment of depression and anxiety (Daley, 2008 Wipfli et al., 2008 Rethorst et al., 2009 Moylan et al., 2013 Rethorst and Trivedi, 2013). These data have prompted some to view exercise as a first line of treatment for mild to moderate depression (Carek et al., 2011). Given the promising data for depression and anxiety, exercise may also prove to be beneficial for the management of bipolar disorder. Specifically, evidence suggests that exercise is neuroprotective at least in part by increasing brain derived neurotrophic factor (BDNF Sylvia et al., 2010). Other mechanisms will be explored, including the genetic expression and endorphin hypothesis.

The aim of this review is to understand the amount of exercise and physical activity currently engaged in by individuals with bipolar disorder. For the purpose of this review, exercise is defined as a conscious, planned decision to move and be physically active, whereas physical activity refers to any movement, including leisure activity, occupational activity, or other activities of daily living (Caspersen et al., 1985 Thompson et al., 2003). A second aim is to evaluate the research on the role of exercise in improving physical (obesity, blood pressure) and mental (symptoms, quality of life) health outcomes in bipolar disorder. Finally, we will discuss the potential mechanisms of how exercise is suspected of improving mood and functioning in bipolar disorder.

J.F. is funded by an MRC Doctoral Training Grant. D.V. is funded by the Research Foundation – Flanders (FWO-Vlaanderen). S.R. is funded by a Society for Mental Health Research Early Career Fellowship (Australia). K.H.N. is funded by NIMH, Janssen, Stanley Medical Research Institute, and Posit Science. B.M. is funded by the German Federal Ministry of Education and Research (BMBF: 01EE1407AE).

We would like to acknowledge the assistance of Prof. David Kimhy (Columbia University) for kindly agreeing to share study data necessary for the meta-analysis. The authors have declared that there are no conflicts of interest in relation to the subject of this study.