Circle Inequalities: Graphing and Writing

Algebra 2 Grades High School 4:43 Video
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Lesson Description

Learn how to write and graph circle inequalities. This lesson covers the equation of a circle, deriving it from the distance formula, and understanding how to shade the appropriate region based on the inequality.

Video Resource

Circle Inequalities (Graphing & Writing)

Mario's Math Tutoring

Duration: 4:43
Watch on YouTube

Key Concepts

  • Equation of a circle: (x - h)² + (y - k)² = r²
  • Relationship between the equation of a circle and the distance formula
  • Circle inequalities and their graphical representation (shading inside or outside the circle)
  • Solid vs. dashed lines in circle inequalities

Learning Objectives

  • Students will be able to write the equation of a circle given its center and radius.
  • Students will be able to graph a circle inequality given its equation.
  • Students will be able to determine the correct shading for a circle inequality (inside or outside the circle).
  • Students will be able to identify the center and radius of a circle from its equation.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the equation of a circle and its relation to the distance formula. Briefly discuss the components of the equation: center (h, k) and radius (r).
  • Video Viewing (10 mins)
    Watch the Mario's Math Tutoring video on Circle Inequalities (Graphing & Writing). Encourage students to take notes on key concepts and examples.
  • Discussion and Examples (15 mins)
    Discuss the video's content. Work through additional examples of writing and graphing circle inequalities. Emphasize the connection between the inequality symbol and the shading direction (inside or outside the circle).
  • Independent Practice (15 mins)
    Students work independently on practice problems involving writing and graphing circle inequalities.
  • Wrap-up and Review (5 mins)
    Review the key concepts and answer any remaining questions. Preview the upcoming lesson on related topics.

Interactive Exercises

  • Graphing Circle Inequalities
    Provide students with a set of circle inequality equations. Have them graph the circles and shade the appropriate regions on graph paper. Check their work for accuracy.
  • Writing Circle Inequalities
    Present students with graphs of circles with shaded regions. Ask them to write the corresponding circle inequality equations. Ensure they correctly identify the center, radius, and inequality symbol.

Discussion Questions

  • How is the equation of a circle derived from the distance formula?
  • What is the difference between a solid and a dashed line when graphing circle inequalities?
  • How does the inequality symbol (>, <, ≥, ≤) determine whether to shade inside or outside the circle?
  • How can you determine the center and radius of a circle from its equation?

Skills Developed

  • Graphing inequalities
  • Algebraic manipulation
  • Problem-solving
  • Visual representation of mathematical concepts

Multiple Choice Questions

Question 1:

The equation of a circle is (x - 2)² + (y + 3)² = 16. What is the center of the circle?

Correct Answer: (2, -3)

Question 2:

The equation of a circle is (x + 1)² + y² = 9. What is the radius of the circle?

Correct Answer: 3

Question 3:

Which inequality represents the region inside the circle (x - 3)² + (y - 1)² = 4?

Correct Answer: (x - 3)² + (y - 1)² < 4

Question 4:

Which inequality represents the region outside the circle (x + 2)² + (y - 4)² = 25, including the circle itself?

Correct Answer: (x + 2)² + (y - 4)² ≥ 25

Question 5:

What type of line is used to graph a circle inequality that does NOT include the points on the circle?

Correct Answer: Dashed

Question 6:

Which point is inside the circle defined by (x-1)^2 + (y+2)^2 < 9?

Correct Answer: (-2, -2)

Question 7:

The center of a circle inequality is at (0,0) and its radius is 5. Which inequality represents all points outside the circle, not including the points on the circle?

Correct Answer: x^2 + y^2 > 25

Question 8:

Which circle inequality would result in a solid line?

Correct Answer: (x-a)^2 + (y-b)^2 ≥ r^2

Question 9:

What is the role of the distance formula in understanding circle equations?

Correct Answer: It provides the foundation for deriving the equation of a circle.

Question 10:

If a point (x, y) satisfies the inequality (x-h)^2 + (y-k)^2 < r^2, where is the point located relative to the circle?

Correct Answer: Inside the circle

Fill in the Blank Questions

Question 1:

The equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) represents the ________.

Correct Answer: center

Question 2:

In the equation of a circle, 'r' represents the ________ of the circle.

Correct Answer: radius

Question 3:

When graphing a circle inequality with a 'greater than or equal to' symbol (≥), the line is ________.

Correct Answer: solid

Question 4:

When graphing a circle inequality representing all points inside the circle, you should ________ the area inside the circle.

Correct Answer: shade

Question 5:

The distance formula is used to ________ the equation of a circle.

Correct Answer: derive

Question 6:

A circle inequality using the less than symbol (<) indicates the solutions are located ________ the circle.

Correct Answer: inside

Question 7:

In the circle inequality (x+5)^2 + (y-2)^2 > 16, the x-coordinate of the center is ________.

Correct Answer: -5

Question 8:

If a circle inequality has a radius of 7, then r^2 is equal to ________.

Correct Answer: 49

Question 9:

In a circle inequality, a ________ line indicates that the points on the circle are NOT included in the solution.

Correct Answer: dashed

Question 10:

The process of visually representing all solutions of a circle inequality on the coordinate plane is known as ________ the circle inequality.

Correct Answer: graphing

Educational Standards

A2.A.1.9:Solve systems of linear inequalities in two variables, with a maximum of three inequalities; graph and interpret the solutions on a coordinate plane. Graphing calculators or other appropriate technology may be used. A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically.

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