Objective 11.1A
Vocabulary to Review
parabola [9.1A]
axis of symmetry [9.1A]
vertex [9.1A]
New Vocabulary
conic sections
y-coordinate of the vertex
New Equations
equations of a parabola:
x- or y-coordinate of the vertex of a parabola:
axis of symmetry for
axis of symmetry for
Discuss the Concepts
1. Describe the vertex and axis of symmetry of a parabola.
2. Explain how, by looking at the equation of a parabola, you can tell whether it opens up or down or opens left or right.
3. If you know the vertex of a parabola, how can you determine the equation of the axis of symmetry of the parabola?
Concept Check
Determine whether the statement is always true, sometimes true, or never true.
1. The graph of a parabola is the graph of a function. Sometimes true
2. The axis of symmetry of a parabola passes through the vertex. Always true
3. The graph of a parabola has two x-intercepts. Sometimes true
4. The graph of a parabola has a minimum value. Sometimes true
5. The axis of symmetry of a parabola is the x-axis or the y-axis. Sometimes true
6. The equation of the axis of symmetry of a parabola is . Sometimes true
Optional Student Activity
Use the vertex and the direction in which the parabola opens to determine the domain and range of the relation.
1. y = x2 – 4x – 2
Domain: {x | x Î real numbers}
Range {y | y ≥ -6}
2. y = x2 – 6x + 1
Domain: {x | x Î real numbers}
Range {y | y ≥ -8}
3. y = -x2 – 2x + 4
Domain: {x | x Î real numbers}
Range {y | y ≤ 5}
4. x = y2 + 6y – 5
Domain: {x | x ≥ -14}
Range: {y | y Î real numbers}
5. x = y2 + 4y – 3
Domain: {x | x ≥ -7}
Range: {y | y Î real numbers}
6. x = -y2 – 2y + 6
Domain: {x | x ≤ 7}
Range: {y | y Î real numbers}
7. x = -y2 – 6y + 2
Domain: {x | x ≤ 11}
Range: {y | y Î real numbers}
Objective 11.2A
New Vocabulary
circle
center of a circle
radius of a circle
Formulas to Review
distance formula
[3.1B]
New Equations
standard form of the equation of a circle:
Discuss the Concepts
1. Explain how to determine the radius and center of the circle given by the equation
2. What do the values of h, k, and r represent in the equation of a circle in standard form?
3. Is the graph of a circle the graph of a function? Why or why not?
4. Explain how the points on the circumference of a circle are related to the center of the circle.
Concept Check
1. Find the equation of the circle that has radius 1, is tangent to both the x- and y-axes, and lies in Quadrant II. (x + 1)2 + (y – 1)2 = 1
2. Find the radius and center of the circle given by the equation Radius: 4; Center: (0, 0)
3. Many communications satellites orbit Earth at an altitude of approximately 22,500 mi above Earth’s surface. Write an equation for the orbit of a communications satellite. Use Earth’s center as the origin and consider the orbit of the satellite circular. (Hint: Earth’s radius is approximately 4000 mi.)
x2 + y2 = (26,500)2
Optional Student Activity
1. Find the equation of the circle that has center (5, -6) and an area of 49 square units.
(x – 5)2 + (y + 6)2 = 49
2. Find the area of the smallest region bounded by the graphs of and
p square units
3. Find the least distance between the graphs of the equations and 3 units
4. The line crosses the circle at points A and B. Find the length of AB. 10 units
Objective 11.2B
New Equations
general form of the equation of a circle:
Vocabulary to Review
completing the square [8.2A]
Discuss the Concepts
1. Explain how to complete the square on
2. Explain how to write the trinomial as the square of a binomial.
3. For the equation explain what it means to group the x terms and group the y terms. What properties of real numbers are used to group these terms?
4. Suppose we have rewritten the equation as (x2 – 6x) + (y2 + 4y) = -4. Explain the next step in writing this equation in standard form.
Concept Check
1. Find the radius of the circle given by the equation 4 units
2. Find the center of the circle given by the equation (4, -3)
Optional Student Activity
Write the equation of the circle in standard form. Then sketch its graph.
1.
2.
Answers to Writing Exercises
21. Attempt to write the equation in standard form.
This is not the equation of a circle because r2 is negative (r2 = -4) and the square of a real number cannot be negative.
Objective 11.3A
New Vocabulary
ellipse
center of an ellipse
New Equations
standard form of the equation of an ellipse with center at the origin: with
x-intercepts: (a, 0), (-a, 0),
y-intercepts: (0, b), (0, -b)
Discuss the Concepts
1. Explain how to determine the domain and range of the relation from its graph. What are the domain and range?
D: {x | -3 ≤ x ≤ 3};
R: {y | -2 ≤ y ≤ 2}
2. Explain how to determine the domain and range of the relation from its equation. What are the domain and range?
D: {x | -5 ≤ x ≤ 5};
R: {y | -4 ≤ y ≤ 4}
Concept Check
1. What are the x-intercepts of the graph of the ellipse ? What are the y-intercepts?
(6, 0), (-6, 0); (0, 3), (0, -3)
2. Write the equation of the ellipse with x-intercepts (7, 0) and (-7, 0) and y-intercepts (0, 2) and (0, -2).
Optional Student Activity
1. The longer axis of symmetry of an ellipse is called the major axis. The shorter axis of symmetry is called the minor axis. The orbit of Halley’s comet is an ellipse with a major axis of approximately 36 AU and a minor axis of approximately 9 AU. (One AU, or one astronomical unit, is approximately 92,960,000 miles, the average distance of Earth from the sun.) Determine an equation for the orbit of Halley’s comet in terms of astronomical units.
2. The orbits of the planets in our solar system are elliptical. The length of the major axis of Mars’s orbit is 3.04 AU. (See Exercise 1 above.) The length of the minor axis is 2.99 AU. Determine an equation for the orbit of Mars.
Objective 11.3B
New Vocabulary
hyperbola
vertices of a hyperbola
axis of symmetry of a hyperbola
asymptote
New Equations
standard form of the equation of a hyperbola with center at the origin:
, with vertices , ;
, with vertices (0, b), (0, -b);
asymptotes of a hyperbola: and
Discuss the Concepts
1. How can you tell from an equation whether its graph will be that of an ellipse or that of a hyperbola?
2. How do you know by looking at the equation of a hyperbola whether it has x-intercepts or y-intercepts?
3. What are the asymptotes of a hyperbola?
4. In addition to the curves presented in this section, how else might the intersection of a plane and a cone be represented?
There are three degenerate conic sections.
(1) The intersection of a plane perpendicular to the axis of the cone and through the vertex of the cone is a point.
(2) The intersection of a plane parallel to the axis of the cone and through the vertex of the cone forms two intersecting straight lines.
(3) The intersection of a plane and the lateral surface of the cone is a line.
Concept Check
Describe the graph of the conic section given by the equation.
1. Parabola that opens up
2. Parabola that opens left
3. Circle with center (4, -2) and radius 1
4. Ellipse with x-intercepts (3, 0) and (-3, 0) and y-intercepts (0, 5) and (0, -5)
5. Hyperbola with x-intercepts (5, 0) and (-5, 0)
6. Hyperbola with y-intercepts (0, 3) and (0, -3)
Optional Student Activity
1. Find the equation of the hyperbola with vertices (0, 4) and (0, -4) and asymptotes and .
2. When are the asymptotes of the graph of perpendicular? When a = b
Answers to Exercises
22. a. ellipse
23. a. ellipse
24. a. hyperbola
25. a. hyperbola
26. a. hyperbola
27. a. hyperbola
Objective 11.4A
Vocabulary to Review
substitution method [4.1B]
addition method [4.2A]
New Vocabulary
nonlinear system of equations
Discuss the Concepts
Determine whether the statement is always true, sometimes true, or never true.
1. It is possible for two ellipses with centers at the origin to intersect in three points. Never true
2. Two circles will intersect in two points. Sometimes true
3. A straight line will intersect a parabola in two points. Sometimes true
4. An ellipse and a circle will intersect in four points. Sometimes true
5. When the graphs of the equations of a system of equations do not intersect, the system has no solution. Always true
6. Two circles will intersect in four points. Never true
Concept Check
1. How do nonlinear systems of equations differ from linear systems of equations?
2. What methods are used to solve nonlinear systems of equations?
3. The difference between two numbers is 3, and the difference between their squares is 27. Find the two numbers. 3 and 6 or -3 and -6
4. Given and , find the value of . 12
5. From the solutions of the system of equations
find the value of y that corresponds to the maximum value of x. -2
Optional Student Activity
Suppose you have been hired to track incoming meteorites and determine whether or not they will strike Earth. The equation of Earth’s circumference is x2 + y2 = 40 You observe a meteorite moving along a path whose equation is 18x – y2 = -144. Will the meteorite hit Earth? No
Objective 11.5A
New Vocabulary
graph of a quadratic inequality in two variables
Discuss the Concepts
1. What does a solid curve on the graph of a nonlinear inequality mean? What does a dashed curve on the graph of a nonlinear inequality mean?
2. When does the solution set of a quadratic inequality in two variables that is bounded by a conic section include the boundary?
3. How can you determine whether (0, 0) is a solution of the inequality x2 + y2 > 4?
4. Does the solution set of a quadratic inequality in two variables that is bounded by a hyperbola include the asymptotes of the hyperbola? Sometimes
Concept Check
For which of the following inequalities does the graph of the solution set not include the origin?
a.
b.
c.
d.
c
Optional Student Activity
Graph xy 1 and y on different coordinate grids. Dividing each side of xy 1 by x yields y , but the graphs are not the same. Explain.
The graphs are shown below. They are not the same because multiplying each side of an inequality by a variable expression does not necessarily produce an equivalent inequality. If x 0, the inequality symbol does not change; if x 0, the inequality symbol is reversed.
Objective 11.5B
Vocabulary to Review
solution set of a system of inequalities [4.5A]
Discuss the Concepts
1. Is the solution set of a nonlinear system of inequalities the union of the solution sets of the individual inequalities? No
2. Explain how you can check that you have shaded the correct region of the plane when graphing the solution set of a system of inequalities.
Concept Check
Which of the following systems of inequalities has a solution set that is the empty set?
a.
b.
c.
d.
d
Optional Student Activity
Graph the solution set.
1.
2.
3.
Answers to Writing Exercises
1. A solid curve is used for the boundaries of inequalities that use ≤ or ≥, and a dashed curve is used for the boundaries of inequalities that use or .
2. Use a point (a, b) to determine which region to shade. If (a, b) is a solution of the inequality, then shade the region of the plane containing (a, b). If (a, b) is not a solution of the inequality, shade the region of the plane that does not contain the point (a, b). Note: (0, 0) is often a good choice for (a, b).
Answers to Focus on Problem Solving: Using a Variety of Problem-Solving Techniques
1. Label the coins 1, 2, 3, 4, 5, 6, 7, and 8. Set coins 1 and 2 aside. Put coins 3, 4, and 5 on one side of the balance and coins 6, 7, and 8 on the other side of the balance. If the scale balances, then coin 1 or coin 2 is the lighter coin. You can determine which one by using the balance a second time.
If the scale does not balance, choose the lighter set of coins. Assume the lighter set includes coins 6, 7, and 8. Choose two of these coins to put on the balance, say coins 6 and 7. If the coins do not balance, you have found the lighter coin.
If coins 6 and 7 are equal in weight, then coin 8 is the lighter coin. Thus the lighter coin can be found in two weighings.
2. Each term, after the first two, is the sum of the two preceding terms. Using that pattern, the next number in the sequence is 21.
3. For example:
Row 1: 6, 7, 2
Row 2: 1, 5, 9
Row 3: 8, 3, 4
4. $48.00
5. a. For 3 g, use 4 g on one side and 1 g plus the 3-gram weight on the other side of the balance.
b. For 7 g, use 8 g on one side and 1 g plus the 7-gram weight on the other side of the balance.
c. For 11 g, use 16 g on one side and 11 g plus the 1-gram and 4-gram weights on the other side of the balance.