Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on conics focuses on parabola, a conic section defined as the set of points equidistant from a focus and a directrix. It explains how to derive the equation of a parabola by using the distance formula. By strategically positioning the vertex at the origin or at an arbitrary point (h, k), the derivations become simpler and more applicable to real-world problems, such as satellite dishes and projectile motions.

Learning Objectives

1

Identify and name the conic sections, with a special emphasis on the parabola.

2

Describe the geometric definition of a parabola using the focus-directrix property.

3

Derive the equation of a parabola with the vertex at the origin and analyze its components.

4

Extend the analysis to parabolas with vertices at arbitrary points (h, k).

5

Solve applied problems involving parabolas using their geometric properties.

Key Concepts

CONCEPT

DEFINITION

Parabola

The set of all points P in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Focus

A fixed point F used in the definition of a parabola, where each point on the parabola is equidistant from F and the directrix.

Directrix

A fixed line D used in the definition of a parabola; the distance from any point on the parabola to D is equal to its distance to the focus.

Vertex

The point of intersection of the parabola with its axis of symmetry, located midway between the focus and the directrix.

Axis of Symmetry

The line that passes through the focus and is perpendicular to the directrix; it divides the parabola into two mirror-image halves.

Example Problems

Example 1

The formula for the distance $d$ from $P_{1}=\left(x_{1}, y_{1}\right)$ to $P_{2}=\left(x_{2}, y_{2}\right)$ is $d=$ ___.

Example 2

To complete the square of $x^{2}-4 x,$ add ____.

Example 3

Use the Square Root Method to find the real solutions of $(x+4)^{2}=9 .

Example 4

The point that is symmetric with respect to the $x$ -axis to the point $(-2,5)$ is ____.

Example 5

To graph $y=(x-3)^{2}+1,$ shift the graph of $y=x^{2}$ to the right ____ units and then ___ 1 unit.
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Step-by-Step Explanations

QUESTION

Given a parabola with vertex at (0,0), focus at (a,0), and directrix x = -a, derive its equation.

STEP-BY-STEP ANSWER:

Step 1: Start with the definition: for any point P(x, y) on the parabola, the distance from P to the focus equals the distance from P to the directrix.
Step 2: Express the distance from P(x, y) to the focus F(a, 0) using the distance formula: √[(x - a)^2 + y^2].
Step 3: Express the distance from P(x, y) to the directrix x = -a as |x + a|.
Step 4: Set these distances equal: √[(x - a)^2 + y^2] = |x + a|.
Step 5: Square both sides to remove the square root, resulting in: (x - a)^2 + y^2 = (x + a)^2.
Step 6: Expand both sides: x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2.
Step 7: Simplify by canceling x^2 and a^2 from both sides, leaving: -2ax + y^2 = 2ax.
Step 8: Rearrange to isolate y^2: y^2 = 4ax.
Final Answer: The equation of the parabola is y^2 = 4ax.

Deriving the Equation of a Parabola (Vertex at Origin, Focus on the x-axis)

QUESTION

How can you modify the standard form of a parabola if the vertex is moved from the origin to a point (h, k)?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the basic equation y^2 = 4ax assumes the vertex is at (0, 0).
Step 2: When the vertex is relocated to (h, k), replace x with (x - h) and y with (y - k) in the equation.
Step 3: For a parabola opening horizontally, the modified equation becomes (y - k)^2 = 4a(x - h).
Step 4: For a parabola that would otherwise open vertically, the form is modified accordingly using (x - h)^2 = 4a(y - k).
Final Answer: The general form for a parabola with vertex (h, k) is (y - k)^2 = 4a(x - h) for horizontal opening or (x - h)^2 = 4a(y - k) for vertical opening.

Analyzing Parabolas with Vertex at (h, k)

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Common Mistakes

  • Mixing up the roles of the focus and the directrix during the derivation.
  • Incorrectly setting up the distance formula, leading to errors in squaring and simplifying the equation.
  • Forgetting to adjust the equation properly when the vertex is not at the origin (failure to substitute x-h and y-k).
  • Overlooking the axis of symmetry and its impact on the orientation of the parabola.