Book cover for Elementary and Intermediate Algebra

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson

ISBN #9781111567682

5th Edition

9,862 Questions

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237,244 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes reflecting on past study habits while mastering the properties of conic sections, particularly circles and parabolas. It covers converting equations from general form to standard form via completing the square, identifying key features such as centers, radii, vertices, and axes of symmetry, and applying these skills to solve practical problems like designing radios and plotting projectile paths. Understanding these foundational techniques is crucial for handling more advanced topics in mathematics.

Learning Objectives

1

Reflect on your study habits and performance to effectively prepare for the next mathematics course.

2

Identify, classify, and describe conic sections—especially circles and parabolas—and understand their real-world applications.

3

Graph circles by converting equations from general form to standard form using techniques like completing the square.

4

Write the equation of a circle given its center and radius and solve application problems involving circles.

5

Convert the general form of a parabola to its standard form and graph parabolas to identify key features such as the vertex, axis of symmetry, and direction of opening.

Key Concepts

CONCEPT

DEFINITION

Conic Section

The curve formed by the intersection of a plane with an infinite right-circular cone, including circles, parabolas, ellipses, and hyperbolas.

Circle

A set of all points in a plane that are a fixed distance (radius) from a fixed point (center). Its standard form is (x – h)² + (y – k)² = r².

Parabola

The set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix). Its standard forms are y = a(x – h)² + k for vertical orientation or x = a(y – k)² + h for horizontal orientation.

Completing the Square

A method used to convert a quadratic equation from general form to standard form by creating a perfect square trinomial.

Standard Form

An equation form that clearly shows the key features of a conic section. For circles, it is (x – h)² + (y – k)² = r²; for parabolas, it is typically y = a(x – h)² + k or x = a(y – k)² + h.

Example Problems

Example 1

Fill in the blanks. The curves formed by the intersection of a plane with an infinite right-circular cone are called _____.

Example 2

Fill in the blanks. Give the name of each conic shown below. ( PICTURE NOT COPY)

Example 3

Fill in the blanks. $A$ _____ is the set of all points in a plane that are a fixed distance from a fixed point called its center. The fixed distance is called the ____.

Example 4

Fill in the blanks. A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed ______.

Example 5

CONCEPTS A. Write the standard form of the equation of a circle. B. Write the standard form of the equation of a circle with the center at the origin.
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Step-by-Step Explanations

QUESTION

How do you convert the general equation x² + y² – 12x + 10y – 20 = 0 to standard form?

STEP-BY-STEP ANSWER:

Step 1: Group x-terms and y-terms: (x² – 12x) + (y² + 10y) = 20.
Step 2: Complete the square for each group. For x: Half of -12 is -6 and (-6)² = 36. For y: Half of 10 is 5 and 5² = 25.
Step 3: Add these squares to both sides: (x² – 12x + 36) + (y² + 10y + 25) = 20 + 36 + 25.
Step 4: Write the left-hand side as perfect squares: (x – 6)² + (y + 5)² = 81.
Final Answer: The standard form is (x – 6)² + (y + 5)² = 81.

Converting a Circle from General Form to Standard Form

QUESTION

How do you write the equation of a circle with center (4, 1) and radius 3?

STEP-BY-STEP ANSWER:

Step 1: Use the standard form equation (x – h)² + (y – k)² = r².
Step 2: Substitute h = 4, k = 1, and r = 3 into the equation.
Step 3: Write the equation as (x – 4)² + (y – 1)² = 9.
Final Answer: The required circle's equation is (x – 4)² + (y – 1)² = 9.

Writing the Equation of a Circle Given Center and Radius

QUESTION

How can you convert the parabola equation x² – 4x + y² – 2y – 11 = 0 to standard form?

STEP-BY-STEP ANSWER:

Step 1: Group x and y terms: (x² – 4x) + (y² – 2y) = 11.
Step 2: Complete the square for x: Half of -4 is -2 and (-2)² = 4. For y: Half of -2 is -1 and (-1)² = 1.
Step 3: Add these constants to both sides to get (x² – 4x + 4) + (y² – 2y + 1) = 11 + 4 + 1.
Step 4: Write as perfect squares: (x – 2)² + (y – 1)² = 16.
Final Answer: The equation in standard form is (x – 2)² + (y – 1)² = 16 (this example, while resembling a circle, emphasizes completing the square; for parabolas, the process is similar but only one variable is squared).

Converting a Parabola to Standard Form

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Common Mistakes

  • Failing to complete the square correctly, often by miscalculating the number to add or subtract.
  • Mixing up the signs when rewriting the standard form of a circle or parabola.
  • Confusing the equation of a circle (which has both x² and y² terms with the same coefficient) with that of a parabola (which has only one squared term).
  • Overlooking the fact that circles do not represent functions, thus causing challenges when graphing with certain graphing calculators.