Lesson Plan: Quadratic Relations – General Form of the Equation of a Parabola

Lesson Objectives:

Understand the general form for vertical and horizontal parabolas.

Common Core Standards:

HSG.GPE.2: Derive the equation of a parabola given a focus and directrix.

Key Vocabulary:

Parabola: A U-shaped curve that is symmetrical and formed by the intersection of a cone with a plane parallel to its side
Multimedia Resource: https://www.media4math.com/library/74573/asset-preview
Horizontal parabola: A parabola that opens to the left or right, with its axis of symmetry parallel to the x-axis
Vertex: The turning point of a parabola, where it changes from increasing to decreasing or vice versa
Multimedia Resource: https://www.media4math.com/library/74587/asset-preview
Focus: A fixed point that helps define the shape of a parabola
Directrix: A fixed line that, together with the focus, determines the shape of a parabola

Warm-up Activities

Choose from these activity types:

Activity 1: Image Analysis

Display images of horizontal parabolas

Horizontal parabola 1: https://www.media4math.com/library/22630/asset-preview
Horizontal parabola 2: https://www.media4math.com/library/22631/asset-preview

Ask students to describe what they observe about the shape, orientation, and key features of these parabolas. Have students compare them to typical quadratic function parabolas.

Activity 2: Desmos

Graph parabolas using Desmos.com

Guide students through graphing several vertical and horizontal parabolas, such as
- y = x²vs x = y²
- y = -x²vs x = -y²
- y = (x - 1)² + 1 vs x = (y - 1)²

For each graph, ask students to identify key features like the vertex, axis of symmetry, and direction of opening.

Activity 3: Verbal Descriptions

Verbal descriptions and sketching

Provide verbal descriptions of parabolas using specific values for a, b, and c in the standard form (y = ax² + bx + c).
Have students quickly sketch these parabolas on graph paper or mini-whiteboards.

Teach (25 minutes)

The Geometric Foundations of a Parabola

A parabola is a geometric figure defined as the locus of points equidistant from a fixed point called the focus and a fixed line called the directrix. This unique property of parabolas makes them central to the study of quadratic equations and their visual representations. Understanding the geometric definition is a foundational step before exploring their algebraic and functional forms. In the illustration below, at any point on the parabola, that point is the same distance away from the focus and the directrix. Note that the distance to the directrix is based on the perpendicular distance.

Visualizing a Parabola on a Coordinate System

On a coordinate system, a parabola is typically represented by the graph of a quadratic equation. For a vertically oriented parabola with the vertex at the origin, the equation is commonly expressed as y = ax². The parabola opens upward if a > 0 and downward if a < 0. Similarly, for a horizontally oriented parabola, the equation takes the form x = ay², opening to the right if a > 0 and to the left if a < 0.

Horizontal vs. Vertical Parabolas

Vertical Parabolas (y = ax²):
- These parabolas are functions, as each x-value corresponds to exactly one y-value. This property satisfies the vertical line test.
Horizontal Parabolas (x = ay²):
- These parabolas are not functions because a single x-value may correspond to multiple y-values. As a result, they fail the vertical line test.

By distinguishing between these orientations and exploring their geometric properties, students build a deeper understanding of how parabolas behave as graphs and as mathematical models.

Focus and Directrix of a Parabola

The geometric definition of a parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). The distance from the vertex to the focus and from the vertex to the directrix is the distance p.

How to Find the Focus and Directrix

Equation: y = ax² + bx + c is rewritten as (x - h)² = 4p(y - k), where (h, k) is the vertex, and (p) is the distance between the vertex and the focus.
Focus: The focus is located at (h, k + p).
Directrix: The directrix is the horizontal line y = k - p.

Rewrite the equation into standard form: Start with the given quadratic equation and manipulate it to fit the standard form of a parabola (x - h)² = 4p(y - k).
Identify the vertex (h, k): The vertex is the midpoint between the focus and the directrix.
Calculate p: From the standard form, 4p is the coefficient in the equation. Solve for p to determine the distance from the vertex to the focus (or the vertex to the directrix).
Determine the focus and directrix: Use the vertex and p to calculate the coordinates of the focus and the equation of the directrix.

Example

Given Equation: y = 2x² - 4x + 5

Complete the square to rewrite in vertex form:
y = 2(x² - 2x) + 5
y = 2((x - 1)² - 1) + 5
y = 2(x - 1)² + 3
Vertex: (1, 3)
Find 4p:
Write the vertex form this way, (x - h)² = 4p(y - k):
(x - 1)² = $ \frac{1}{2} $(y - 3)
(x - 1)² = 4p(y - 3)

This means
4p = $\frac{1}{2}$
$p = \frac{1}{8}$.
Calculate Focus and Directrix:
Focus: (1, 3 + $ \frac{1}{8} $) = (1, $ \frac{25}{8} $)
Directrix: $y = 3 - \frac{1}{8}$ = $ \frac{23}{8} $

Slide Show

Use this slide show to go over the definition of a parabola. This slide show also includes worked-out examples and a Desmos template:
https://www.media4math.com/library/slideshow/student-tutorial-equations-parabolas

The general form equation for a vertical parabola is: y = a(x - h)² + k

For a horizontal parabola: x = a(y - k)² + h

Where (h, k) is the vertex of the parabola, and 'a' determines the direction and steepness of the parabola.

Study the following examples to learn more about parabolas.

Example 1: Vertical Parabola

Graph the parabola with the equation y = 2(x - 3)² - 4. Find the vertex, focus, and directrix.

Identify the vertex: (h, k) = (3, -4)
Rewrite the vertex form to find the vertex:

(x - 3)² = $ \frac{1}{2} $(y + 4)

(x - 3)² = 4p(y + 4)

Calculate the value of p:

4p = $ \frac{1}{2} $

p = $ \frac{1}{8} $

Find the focus and directrix:

Focus: (3, -4 + $ \frac{1}{8} $) = (3, -3$ \frac{7}{8} $)
Directrix: $y = -4 - \frac{1}{8}$ = -4$ \frac{1}{8} $

Example 2: Horizontal Parabola

Problem: Given the equation of the parabola:

y² - 6y + 4x - 5 = 0

Find the vertex, focus, and directrix.

Step 1: Rewrite the Equation in Standard Form

Starting with the equation:

y² - 6y + 4x - 5 = 0

Group the terms involving y:

y² - 6y = -4x + 5

Complete the square:

Take half the coefficient of y (-6), square it, and add to both sides:
(-6 / 2)² = 9
Add 9 to both sides:
y² - 6y + 9 = -4x + 5 + 9

Simplify:

(y - 3)² = -4x + 14

Rearrange to standard form:

(y - 3)² = -4(x - 3.5)

Step 2: Identify the Vertex

From the standard form:

(y - k)² = 4p(x - h)

Vertex: (3.5, 3)

Step 3: Find the Focus

From 4p = -4, solve for p:

p = -1

Focus: (2.5, 3)

Step 4: Determine the Directrix

The directrix is a vertical line located p units to the right of the vertex:

Directrix: x = 4.5

Example 3: Comet Approaching the Sun

A comet's path as it approaches the sun can be modeled by a horizontal parabola. Let the sun be at the origin (0, 0), and the comet's path be described by the equation: x = $ \frac{1}{16} $(y - 4)² - 9.

Identify the vertex: (h, k) = (-9, 4). Note: This is a horizontal parabola.
The parabola opens to the right (a > 0), representing the comet's approach to the sun.
Graph the parabola, noting that the vertex represents the comet's closest approach to the sun
Calculate p:

(y - 4)² = $ \frac{1}{16} $(x + 9)

(x - 3)² = 4p(y + 4)

4p = $ \frac{1}{16} $

p = $ \frac{1}{64} $

Calculate the focus and directrix:
- focus: (-9 + $ \frac{1}{64} $, 4)
- directrix: x = -9 - $ \frac{1}{64} $

Summary: This real-world application demonstrates how horizontal parabolas can model physical phenomena. The equation allows us to determine the comet's closest approach to the sun (-9 units from the origin) and its trajectory as it moves through space.

Review (10 minutes)

Key Vocabulary

Parabola: A U-shaped curve that is symmetrical and formed by the intersection of a cone with a plane parallel to its side.
Vertical Parabola: A parabola that opens upward or downward, with its axis of symmetry parallel to the y-axis.
Horizontal Parabola: A parabola that opens to the left or right, with its axis of symmetry parallel to the x-axis.
Vertex: The turning point of a parabola, where it changes from increasing to decreasing or vice versa.
Focus: A fixed point that helps define the shape of a parabola.
Directrix: A fixed line that, together with the focus, determines the shape of a parabola.

Example 1: Finding the Vertex, Focus, and Directrix of a Vertical Parabola

Given the quadratic equation of a vertical parabola:

(x - 2)² = 8(y - 1)

Identify the Standard Form: The equation is in the form (x - h)² = 4p(y - k), where (h, k) is the vertex and p determines the distance to the focus and directrix. Here, (h, k) = (2, 1), so the vertex is at (2, 1).
Determine p: From the equation, 4p = 8, so p = 2.
Find the Focus: The focus is located p units above the vertex for a vertical parabola. Since the vertex is at (2, 1), the focus is at (2, 3).
Find the Directrix: The directrix is a horizontal line p units below the vertex. Thus, the directrix is y = -1.

Summary:

Vertex: (2, 1)
Focus: (2, 3)
Directrix: y = -1

Example 2: Finding the Vertex, Focus, and Directrix of a Horizontal Parabola

Given the quadratic equation of a horizontal parabola:

(y + 3)² = -12(x - 4)

Identify the Standard Form: The equation is in the form (y - k)² = 4p(x - h), where (h, k) is the vertex and p determines the distance to the focus and directrix. Here, (h, k) = (4, -3), so the vertex is at (4, -3).
Determine p: From the equation, 4p = -12, so p = -3.
Find the Focus: The focus is located p units to the left of the vertex for a horizontal parabola (negative p indicates the left direction). Since the vertex is at (4, -3), the focus is at (1, -3).
Find the Directrix: The directrix is a vertical line p units to the right of the vertex. Thus, the directrix is x = 7.

Summary:

Vertex: (4, -3)
Focus: (1, -3)
Directrix: x = 7

Additional Multimedia Resources

For further exploration, watch this video on applications of parabolic equations: https://www.media4math.com/library/39593/asset-preview

Quiz

Directions: Answer the following questions based on our lesson on the general form of parabola equations. Show your work where applicable.

Write the general form equation for a vertical parabola.
What does the value of 'a' in the general form equation indicate about a parabola?
Graph the parabola with the equation y = -2(x + 1)² + 3.
Identify the vertex and direction of opening for the parabola x = (y - 2)² - 4.
A projectile's path is modeled by the equation y = -0.05(x - 40)² + 50. What is the maximum height reached by the projectile?
Write the equation of a horizontal parabola with vertex (3, -2) that opens to the left.
How does the general form equation of a parabola relate to its focus and directrix?
Sketch a horizontal parabola that opens to the right and has a vertex at (-1, 2).
Explain the difference between the equations of vertical and horizontal parabolas.
A satellite dish is shaped like a parabola with the equation y = 0.1(x - 5)². Where should the receiver be placed to best capture signals?

Answer Key

y = a(x - h)² + k
The value of 'a' indicates the direction of opening (up if a > 0, down if a < 0) and the steepness of the parabola.
Graph should show a parabola opening downward with vertex at (-1, 3)
Vertex: (-4, 2), opens to the right
50 units (the k-value in the equation)
x = -a(y + 2)² + 3, where a is any positive number
The general form equation defines the set of points equidistant from the focus and directrix.
Sketch should show a parabola opening to the right with vertex at (-1, 2)
Vertical parabolas use y = a(x - h)² + k, while horizontal parabolas use x = a(y - k)² + h.
Vertex: $(5, 0)$ $($ $5$ $,$ $0$ $)$ . Focus: One-fourth of $1/0.1 = 2.5$ $1/0.1$ $=$ $2.5$ units above vertex at $(5, 1.25)$ $($ $5$ $,$ $1.25$ $)$ .