Pre-Calculus Unit 4 Test - Free Response

Instructions:

- A handheld calculator is allowed for this assignment.
- Show all of your work, even if the question may not explicitly remind you to do so.
- Clearly label any functions, graphs, tables, or other objects that you use.
- Justifications require that you give mathematical reasons and verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied.
- Your work will be scored on the correctness and completeness of your methods as well as your answers.
- Answers without supporting work or explanation will not receive credit.
- Work should be handwritten only.
- Unless otherwise specified, answers (numeric or algebraic) need not be simplified.
- If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.
- Unless otherwise specified, the domain of a function is assumed to be the set of all real numbers for which the value of the function is a real number.

Questions:

1. What is the difference between the equations for parabolas, ellipses, circles, and hyperbolas? (10 points)

2. Write the standard equations of a hyperbola with a horizontal transverse axis and a hyperbola with a vertical transverse axis. (10 points)

Question

29-07-2024
Mathematics

Answered

Pre-Calculus Unit 4 Test - Free Response

Instructions:

- A handheld calculator is allowed for this assignment.
- Show all of your work, even if the question may not explicitly remind you to do so.
- Clearly label any functions, graphs, tables, or other objects that you use.
- Justifications require that you give mathematical reasons and verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied.
- Your work will be scored on the correctness and completeness of your methods as well as your answers.
- Answers without supporting work or explanation will not receive credit.
- Work should be handwritten only.
- Unless otherwise specified, answers (numeric or algebraic) need not be simplified.
- If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.
- Unless otherwise specified, the domain of a function is assumed to be the set of all real numbers for which the value of the function is a real number.

Questions:

1. What is the difference between the equations for parabolas, ellipses, circles, and hyperbolas? (10 points)

2. Write the standard equations of a hyperbola with a horizontal transverse axis and a hyperbola with a vertical transverse axis. (10 points)

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-29T10:56:01-04:00

Final answer:

The answer explains the differences between parabolas, ellipses, circles, and hyperbolas, and provides the standard equations for hyperbolas with different orientations. It also outlines the process to find the Equations to the axes, asymptotes, and conjugate hyperbola of a hyperbola given a general form of the equation.

Explanation:

The difference between the equations for parabolas, ellipses, circles, and hyperbolas lies in their algebraic expressions and geometric properties. For example, parabolas have equations in the form y=ax²+bx+c, ellipses have equations in the form x²/a² + y²/b² = 1 or y²/a² + x²/b² = 1, circles have equations like (x-h)² + (y-k)² = r², and hyperbolas have equations in the form x²/a² - y²/b² = 1 or y²/b² - x²/a² = 1.

The standard equations of hyperbolas with different orientations are: Horizontal transverse axis: (x-h)²/a² - (y-k)²/b² = 1, and Vertical transverse axis: (y-k)²/a² - (x-h)²/b² = 1. The axes of a hyperbola, asymptotes, and conjugate hyperbola can be derived by manipulating the given general form of the hyperbola equation.

To find the Equations to the axes, asymptotes, and conjugate hyperbola of a hyperbola given by ax² +2hxy+by2² +2gx+2fy+c=0, you can refer the hyperbola to new coordinates, then manipulate the equation to find the desired forms. For example, replacing x with x' + xo and y with y' + yo can help simplify the equation to derive the Equations to the axes, asymptotes, and conjugate hyperbola.

Learn more about Equations for Conic Sections here:

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