In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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0
Prerequisites and Review
0.1
Real Numbers
0.2
Integer Exponents and Scientific Notation
0.3
Polynomials: Basic Operations
0.4
Factoring Polynomials
0.5
Rational Expressions
0.6
Rational Exponents and Radicals
0.7
Complex Numbers
1
Equations and Inequalities
1.1
Linear Equations
1.2
Applications Involving Linear Equations
1.3
Quadratic Equations
1.4
Other Types of Equations
1.5
Other Types of Equations
1.6
Polynomial and Rational Inequalities
1.7
Absolute Value Equations and Inequalities
2
Graphs
2.1
Basic Tools: Cartesian Plane, Distance, and Midpoint
2.2
Graphing Equations: Point-Plotting, Intercepts, and Symmetry
2.3
Lines
2.4
Circles
2.5
Linear Regression: Best Fit
3
Functions and Their Graphs
3.1
Functions
3.2
Graphs of Functions
3.3
Graphing Techniques: Transformations
3.4
Operations on Functions and Composition of Functions
3.5
One-to-One Functions and Inverse Functions
3.6
Modeling Functions Using Variation
4
Polynomial and Rational Functions
4.1
Quadratic Functions
4.2
Polynomial Functions of Higher Degree
4.3
Dividing Polynomials: Long Division and Synthetic Division
4.4
The Real Zeros of a Polynomial Function
4.5
Complex Zeros: The Fundamental Theorem of Algebra
4.6
Rational Functions
5
Exponential and Logarithmic Functions
5.1
Exponential Functions and Their Graphs
5.2
Logarithmic Functions and Their Graphs
5.3
Properties of Logarithms
5.4
Exponential and Logarithmic Equations
5.5
Exponential and Logarithmic Models
6
Trigonometric Functions
6.1
Angles, Degrees, and Triangles
6.2
Definition 1 of Trigonometric Functions: Right Triangle Ratios
6.3
Applications of Right Triangle Trigonometry: Solving Right Triangles
6.4
Definition 2 of Trigonometric Functions: Cartesian Plane
6.5
Trigonometric Functions of Nonacute Angles
6.6
Radian Measure and Applications
6.7
Definition 3 of Trigonometric Functions: Unit Circle Approach
6.8
Graphs of Sine and Cosine Functions
6.9
Graphs of Other Trigonometric Functions
7
Analytic Trigonometry
7.1
Basic Trigonometric Identities
7.2
Verifying Trigonometric Identities
7.3
Sum and Difference Identities
7.4
Double-Angle Identities
7.5
Half-Angle Identities
7.6
Product-to-Sum and Sum-to-Product Identities
7.7
Inverse Trigonometric Functions
7.8
Trigonometric Equations
8
Additional Topics in Trigonometry
8.1
Oblique Triangles and the Law of Sines
8.2
The Law of Cosines
8.3
The Area of a Triangle
8.4
Vectors
8.5
The Dot Product
8.6
Polar (Trigonometric) Form of Complex Numbers
8.7
Polar Equations and Graphs
8.8
Polar Equations and Graphs
9
Systems of Linear Equations and Inequalities
9.1
Systems of Linear Equations in Two Variables
9.2
Systems of Linear Equations in Three Variables
9.3
Partial Fractions
9.4
Systems of Linear Inequalities in Two Variables
9.5
The Linear Programming Model
10
Matrices
10.1
Matrices and Systems of Linear Equations
10.2
Matrix Algebra
10.3
Matrix Equations; The Inverse of a Square Matrix
10.4
The Determinant of a Square Matrix and Cramers Rule
11
Analytic Geometry and Systems of Nonlinear Equations and Inequalities
11.1
Conic Basics
11.2
The Parabola
11.3
The Ellipse
11.4
The Hyperbola
11.5
Systems of Nonlinear Equations
11.6
Systems of Nonlinear Inequalities
11.7
Rotation of Axes
11.8
Polar Equations of Conics
11.9
Parametric Equations and Graphs
12
Sequences, Series, and Probability
12.1
Sequences and Series
12.2
Arithmetic Sequences and Series
12.3
Geometric Sequences and Series
12.4
Mathematical Induction
12.5
The Binomial Theorem
12.6
Counting, Permutations, and Combinations
12.7
Probability
Textbook Solutions for Algebra and Trigonometry,
Chapter 11.7 Problem 9
Question
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
Solution
The first step in solving 11.7 problem number 9 trying to solve the problem we have to refer to the textbook question: In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
From the textbook chapter Rotation of Axes you will find a few key concepts needed to solve this.
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Title
Algebra and Trigonometry, 3
Author
James Stewart, Lothar Redlin, Saleem Watson
ISBN
9780840068132
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform
Chapter 11.7 textbook questions
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Chapter 11: Problem 1 Algebra and Trigonometry, 3
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Chapter 11: Problem 2 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 3 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 4 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 5 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 6 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 7 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 8 Algebra and Trigonometry, 3
In Exercises 18, the coordinates of a point in the xy-coordinate system are given. Assuming that the XY-axes are found by rotating the xy-axes by an angle , find the corresponding coordinates for the point in the XY-system.
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Chapter 11: Problem 9 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 10 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 11 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 12 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 13 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 14 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 15 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 16 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 17 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 18 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 19 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 20 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 21 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 22 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 23 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 24 Algebra and Trigonometry, 3
In Exercises 924, (a) identify the type of conic from the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY-term) by rotating the x- and y-axes by an angle to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).
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Chapter 11: Problem 25 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 26 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 27 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 28 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 29 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 30 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 31 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 32 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 33 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 34 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 35 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 36 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 37 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 38 Algebra and Trigonometry, 3
In Exercises 2538, determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY-term.
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Chapter 11: Problem 39 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 40 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 41 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 42 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 43 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 44 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 45 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 46 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 47 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 48 Algebra and Trigonometry, 3
In Exercises 3948, graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy-term.)
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Chapter 11: Problem 49 Algebra and Trigonometry, 3
The graph of the equation where k is any positive constant less than 6, is an ellipse
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Chapter 11: Problem 50 Algebra and Trigonometry, 3
The graph of the equation where k is any constant greater than 6, is a parabola.
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Chapter 11: Problem 51 Algebra and Trigonometry, 3
The reciprocal function is a rotated hyperbola.
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Chapter 11: Problem 52 Algebra and Trigonometry, 3
The equation can be transformed into the equation X2 + Y2 = 9.
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Chapter 11: Problem 53 Algebra and Trigonometry, 3
Determine the equation in X and Y that corresponds to when the axes are rotated through
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Chapter 11: Problem 54 Algebra and Trigonometry, 3
Determine the equation in X and Y that corresponds to when the axes are rotated through
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Chapter 11: Problem 55 Algebra and Trigonometry, 3
Identify the conic section with equation for various values of a.
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Chapter 11: Problem 56 Algebra and Trigonometry, 3
Identify the conic section with equation for various values of a.
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Chapter 11: Problem 57 Algebra and Trigonometry, 3
Use a graphing utility to explore the second-degree equation for the following values of D, E, and F: a. , b. Show the angle of rotation to the nearest degree. Explain the differences.
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Chapter 11: Problem 58 Algebra and Trigonometry, 3
Use a graphing utility to explore the second-degree equation for the following values of D, E, and F: a. , b. Show the angle of rotation to the nearest degree. Explain the differences.
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Chapter 11: Problem 59 Algebra and Trigonometry, 3
Use a graphing utility to explore the second-degree equation for the following values of D, E, and F: a. , b. Show the angle of rotation to the nearest degree. Explain the differences.
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Chapter 11: Problem 60 Algebra and Trigonometry, 3
Use a graphing utility to explore the second-degree equation for the following values of D, E, and F: a. , b. Show the angle of rotation to the nearest degree. Explain the differences.
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Chapter 11: Problem 61 Algebra and Trigonometry, 3
Use a graphing utility to explore the second-degree equation for the following values of A, B, and C: a. b. c. Show the angle of rotation to the nearest degree. Explain the differences
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Chapter 11: Problem 62 Algebra and Trigonometry, 3
Use a graphing utility to explore the second-degree equation for the following values of A, B, and C: a. b. Show the angle of rotation to the nearest degree. Explain the differences.
Read more