 # K12MATH011: Algebra II

Unit 8: Conic Sections   In this unit, you will be introduced to the connection between geometry and algebra, called conic sections, which will play an important role when you begin the study of calculus. First, you will be introduced to some basics, such as midpoint and distance formulas. Then, you will begin learning certain geometric forms in algebraic expression. You will learn to identify common forms, rewrite equations into common forms, and extract important information, given the rewritten equations. One of the most important methods learned in this section will be completion of squares.

Unit 8 Time Advisory
Completing this unit should take approximately 21 hours and 15 minutes.

☐    Subunit 8.1: 3 hours and 15 minutes

☐    Subunit 8.2: 4 hours and 45 minutes

☐    Subunit 8.3: 4 hours

☐    Subunit 8.4: 4 hours and 15 minutes

☐    Subunit 8.5: 5 hours

Unit8 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Find the midpoint of a line segment between two points. - Calculate the distance between any two points. - Identify whether an equation expresses a parabola, a circle, an ellipse, or a hyperbola. - Rewrite an equation into the standard form of a parabola, a circle, an ellipse, or a hyperbola. - Graph a parabola, a circle, an ellipse, or a hyperbola.

8.1 Midpoint and Distance Formulas   Showing applications to geometry from algebra is a common procedure. Two of the most common applications are midpoint and distance formulas. Midpoint coordinates are nothing but the average of the x and y components of the two points on the line segment equal distance from the center between them. And the distance between two points is the hypotenuse of the right triangle formed by the vertical and horizontal distances between the x and y components of those points.

• Explanation: HippoCampus: “Pythagorean Theorem” Link: HippoCampus: “Pythagorean Theorem” (Flash)

Instructions: On this webpage, select “Geometry” from the “Presentations” section of the menu on the left (it’s under the subheading of “Art of Problem Solving Collection”). Then scroll down to “Right Triangles and Quadrilaterals” in the secondary menu and select “Using the Pythagorean Theorem Part 1.” In this section, you will view a video on using the Pythagorean Theorem to find the distance between two points on a plane. This theorem is one of the most used in mathematics. Then click on “Using the Pythagorean Theorem Part 2” to watch the second part of this video.

Watching the video and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

Instructions: This video demonstrates using the Pythagorean Theorem to find the distance between two points on a plane. This shows a practical application from geometry and algebra and can be used to find the distance between any two points, if a coordinate grid can be created that contains both.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

Instructions: This video demonstrates using the average of the x and y coordinates to find the midpoint between two points on a plane. This shows a practical application from geometry and algebra and can be used to find the halfway point between any two points, if a coordinate grid can be created that contains both.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Distance Formula” Link: Khan Academy’s “Distance Formula” (HTML)

Instructions: This is a self-quiz on the distance formula (which uses the Pythagorean Theorem). Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Midpoint Formula” Link: Khan Academy’s “Midpoint Formula” (HTML)

Instructions: This is a self-quiz on the midpoint formula. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

8.2 Parabolas   Parabolas are formed from quadratic equations and are conic sections that have the property that all lines that reflect off the curve are directed at a focus. Parabolic mirrors may be used to cook, as a result of concentrating sunlight at a single point, and parabolic antennas both send and receive signals that can be directed from or to a particular point within the antenna – the focus.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 3: Parabolas” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 3: Parabolas” (PDF)

Instructions: Read pages 505–511 for information on parabolas. This section demonstrates how the geometry of parabolas, which includes the focus, is connected to algebra.

Reading the article and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy: “Parabola Focus and Directrix 1” Link: Khan Academy: “Parabola Focus and Directrix 1” (YouTube)

Instructions: This video covers the relationship of parabolas to their algebraic equations. This shows a practical connection from geometry and algebra and can be used to find a quadratic equation given the focus and directrix of a parabola. This provides another use of the Pythagorean Theorem.

Watching the video and writing notes should take approximately 30 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy: “Focus and Directrix of a Parabola 2” Link: Khan Academy: “Focus and Directrix of a Parabola 2” (YouTube)

Instructions: This video continues the discussion of the relationship of parabolas to their algebraic equations. This shows a practical connection from geometry and algebra and can be used to find a quadratic equation given the focus and directrix of a parabola. Combined with the reading, you should develop a good understanding of parabolas.

Watching the video and writing notes should take approximately 30 minutes.

Standards Addressed (Common Core):

• Web Media: GeoGebra: “Parabola Simulation” Link: GeoGebra: “Parabola Simulation” (HTML)

Instructions: This interactive graph demonstrates how the focus and directrix work together to shape the parabola.

Practicing with this application and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Web Media: GeoGebra: “Parabola Parameters” Link: GeoGebra: “Parabola Parameters” (HTML)

Instructions: This interactive graph demonstrates how the graph of a parabola changes as you change values in transformational form (f(x) = a(x+h)2 + k). Also given is the standard form so that you may compare the two as changes are made.

Practicing with this application and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Parabola Intuition 1” Link: Khan Academy’s “Parabola Intuition 1” (HTML)

Instructions: This is a short self-assessment on parabolas.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

8.3 Circles   Circles are conic sections formed by all points a given distance from a center point and the formula for them is an application of the Pythagorean Theorem: x2 + y2 = r2 or x2/r2 + y2/r2 = 1. r is called the “radius” of the circle.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 2: Circles” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 2: Circles” (PDF)

Instructions: This section contains information on the circle, including completing the square to rewrite circle equations. Read pages 498–502, and then try the problems at the end of the section. Answers are on the pages that follow.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy: “Intro to Circles” Link: Khan Academy: “Intro to Circles” (YouTube)

Instructions: This video describes circles relative to their algebraic equations. This shows a practical connection from geometry and algebra and can be used to graph a circle given its equation. Once again, the Pythagorean Theorem is used to convert a concept of a radius from the center of the circle into an equation.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Graphing Circles” Link: Khan Academy’s “Graphing Circles” (HTML)

Instructions: This is a short self-assessment on graphing circles. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Equation of a Circle in Factored Form” Link: Khan Academy’s “Equation of a Circle in Factored Form” (HTML)

Instructions: This is a short self-assessment on circle equations in factored (transformational) form. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Equation of a Circle in Non-Factored Form” Link: Khan Academy’s “Equation of a Circle in Non-Factored Form” (HTML)

Instructions: This is a short self-assessment on circle equations in nonfactored (standard) form. Note: To find the radius and center, you will have to apply the process of completing the square. Note also that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Graphing Circles 2” Link: Khan Academy’s “Graphing Circles 2” (HTML)

Instructions: This is a short self-assessment on graphing circles in nonfactored (standard) form. Note: to find the radius and center, you will have to apply the process of completing the square. Note also that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

8.4 Ellipses   Ellipses are a general breed of conic sections to which circles are a special case. Of the form x2/a2 + y2/b2 = 1, they are oblong-shaped geometric figures formed by points at a determined distance from two foci, reflected across an axis. When a = b, instead of two foci, we get one center point, and we have a circle. Ellipses are bounded by the sides of a “box,” whose sides are formed a and b units from the center. The long side is called the major axis and the shorter side the minor axis.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 4: Ellipses” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 4: Ellipses” (PDF)

Instructions: This section contains information on ellipses, including completing the square to rewrite ellipse equations. Read pages 516–524, and then try the problems at the end of the section. Answers are on the pages that follow. After reading the material and completing the problems, write a one-page essay showing how circles and ellipses are related. Is it possible to write a circle’s equation in the form of an ellipse?

Completing this activity should take approximately 1 hour and 30 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy: “Intro to Ellipses” Link: Khan Academy: “Intro to Ellipses” (YouTube)

Instructions: This video describes ellipses relative to their algebraic equations, including how to complete the square to place them in transformational form. This shows a practical connection from geometry and algebra and can be used to graph an ellipse given its equation. Once again, the Pythagorean Theorem is used to convert a concept of a radius from the center of the ellipse into an equation.
Watching the video and writing notes should take approximately 30 minutes.

Standards Addressed (Common Core):

• Web Media: GeoGebra: “Definition of an Ellipse” Link: GeoGebra: “Definition of an Ellipse” (HTML)

Instructions: This interactive graph shows how an ellipse would have been constructed from thread, set foci, and a drawing implement. It also shows how an equation of a basic ellipse in transformational form changes if you move the foci. Note that the closer the foci are placed together, the more circular the construction becomes.

Practicing with this application and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: CK-12’s “Equation of an Ellipse” Link: CK-12’s “Equation of an Ellipse” (HTML and YouTube)

Instructions: Read the passage and watch the video. Scroll down to complete the guided practice and practice problems.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

8.5 Hyperbolas   Hyperbolas are conic sections very similar to ellipses, in that they have major and minor axes and are generally in the form of x2/a2– y2/b2= 1. Instead of being bounded by a and b, a and b form a box through which the asymptotes for the graph are diagonals of the “box,” whose sides are formed by a and b.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 5: Hyperbolas” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 7, Section 5: Hyperbolas” (PDF)

Instructions: This section contains information on identifying and constructing hyperbolas. This includes completing the square to place a hyperbola in transformational form. Read pages 531–540, and then try the problems at the end of the section. Answers are on the pages that follow. After reading the article and writing notes, write a one-page essay on the relationship between ellipses and hyperbolas. How are the two related? How are they different? What common information helps in graphing them?

Completing this activity should take approximately 1 hour and 30 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s: “Intro to Hyperbolas” Link: Khan Academy’s: “Intro to Hyperbolas” (YouTube)

Instructions: This video describes hyperbolas relative to their algebraic equations. This shows a practical connection from geometry and algebra and can be used to graph a hyperbola given its equation. A brief connection to ellipses is made.

Watching the video and writing notes should take approximately 30 minutes.

Standards Addressed (Common Core):

• Web Media: GeoGebra: “Graphing a Hyperbola” Link: GeoGebra: “Graphing a Hyperbola” (HTML)

Instructions: This interactive graph allows you to see how the graph (and its equation) changes if one focus or a point on a curve is moved. This is good practice toward linking the graph with the equation in your mind. It also reemphasizes the fact that a conic section is constructed from a relationship among the foci, the center, and some point.

Practicing with this application and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Checkpoint: Mathematics Vision Project “8.2 Getting Centered” Link: Mathematics Vision Project “8.2 Getting Centered” (PDF)

Instructions: This is another assessment on understanding circles. Scroll to page 7. Complete the task and all of the questions on pages 7-9.

Completing this assessment should take approximately 30 minutes.

Standards Addressed (Common Core):

• Checkpoint: Mathematics Vision Project “8.6 Circles and Other Conics” Link: Mathematics Vision Project “8.6 Circles and Other Conics” (PDF)

Instructions: This is another assessment on understanding conic sections. Scroll to page 32. Complete the “Ready, Set, Go!” questions 1-15.

Completing this assessment should take approximately 45 minutes.

Standards Addressed (Common Core):

• Checkpoint: Khan Academy’s “Equation of an Ellipse” Link: Khan Academy’s “Equation of an Ellipse” (HTML)

Instructions: This is an assessment on understanding ellipses. Work all problems. Answers are available for viewing by a button on each problem. Hints are available, if needed.

Completing this assessment should take approximately 15 minutes.

Standards Addressed (Common Core):

• Checkpoint: Khan Academy’s “Parabola Intuition 2” Link: Khan Academy’s “Parabola Intuition 2” (HTML)

Instructions: This is an assessment on understanding parabolas. Work all problems. Answers are available for viewing by a button on each problem.

Completing this assessment should take approximately 15 minutes.

Standards Addressed (Common Core):