Graphing Systems of Quadratic Inequalities

Algebra 2 Grades High School 4:14 Video
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Lesson Description

This lesson plan teaches students how to graph systems of quadratic inequalities, focusing on identifying key features of parabolas and shading the correct regions to represent solutions.

Video Resource

Graphing System of Quadratic Inequalities Example 1

Mario's Math Tutoring

Duration: 4:14
Watch on YouTube

Key Concepts

  • Vertex form of a quadratic equation
  • Intercept form of a quadratic equation
  • Shading to represent inequality solutions
  • Solid vs. Dashed Lines

Learning Objectives

  • Students will be able to graph quadratic inequalities in vertex and intercept form.
  • Students will be able to determine the correct region to shade for a quadratic inequality.
  • Students will be able to graph a system of quadratic inequalities and identify the solution region.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic concepts of quadratic equations and inequalities. Briefly discuss the standard forms of quadratic equations (vertex and intercept form) and their corresponding graphs (parabolas). Mention how inequalities differ from equations, setting the stage for graphing inequalities.
  • Video Viewing (5 mins)
    Play the video 'Graphing System of Quadratic Inequalities Example 1' by Mario's Math Tutoring (https://www.youtube.com/watch?v=FUUKuZ4jv08). Encourage students to take notes on the steps involved in graphing quadratic inequalities.
  • Vertex Form Example (10 mins)
    Walk through the example from the video (y ≤ -2(x-2)^2 + 4) step-by-step. Emphasize identifying the vertex (2,4), determining whether the parabola opens upwards or downwards (downwards because of the negative coefficient), and deciding whether the line is solid or dashed (solid because of the 'less than or equal to' sign). Demonstrate how to choose a test point to determine the correct region to shade.
  • Intercept Form Example (10 mins)
    Walk through the example from the video (y > (x-2)(x-4)). Emphasize identifying the x-intercepts (2 and 4), determining whether the parabola opens upwards or downwards (upwards because of the positive coefficient), and deciding whether the line is solid or dashed (dashed because of the 'greater than' sign). Demonstrate how to choose a test point to determine the correct region to shade.
  • System of Inequalities (10 mins)
    Explain how to graph both inequalities on the same coordinate plane. The solution to the system is the region where the shaded areas of both inequalities overlap. Emphasize the importance of accurately graphing each inequality and shading the correct regions.
  • Summary and Practice (5 mins)
    Summarize the key steps for graphing systems of quadratic inequalities. Assign practice problems for students to complete individually or in pairs. Review the shading tips from the video.

Interactive Exercises

  • Graphing Challenge
    Provide students with a worksheet containing several systems of quadratic inequalities. Have them graph each system and identify the solution region. Use graphing calculators or online tools to verify their answers.

Discussion Questions

  • How does the vertex form of a quadratic equation help us graph the parabola?
  • Why do we use a test point when graphing inequalities?
  • How does the inequality symbol affect the type of line we draw (solid vs. dashed)?
  • How does the coefficient of the x^2 term affect the way the parabola opens?

Skills Developed

  • Graphing quadratic functions
  • Solving and graphing inequalities
  • Interpreting mathematical models
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the vertex of the parabola represented by the equation y = (x-3)^2 + 2?

Correct Answer: (3, 2)

Question 2:

If the inequality symbol is '<' or '>', which type of line should be used when graphing the inequality?

Correct Answer: Dashed line

Question 3:

Which point is best to use as a test point?

Correct Answer: (0,0)

Question 4:

Which direction does the graph of y=-x^2 open?

Correct Answer: Down

Question 5:

What form is the equation y = a(x-h)^2 + k in?

Correct Answer: Vertex

Question 6:

When graphing y > (x-a)(x-b), what are 'a' and 'b'?

Correct Answer: x-intercepts

Question 7:

Which action do you perform after graphing the inequality?

Correct Answer: Shade

Question 8:

The area in a system of inequalities that satisfies all inequalities is called the:

Correct Answer: Solution Region

Question 9:

If the test point (0,0) satisfies the inequality, where do you shade?

Correct Answer: Toward (0,0)

Question 10:

Which equation has a vertex of (2, -3)?

Correct Answer: y = (x-2)^2 - 3

Fill in the Blank Questions

Question 1:

The highest or lowest point on a parabola is called the ________.

Correct Answer: vertex

Question 2:

When graphing an inequality, if the symbol is ≤ or ≥, use a ________ line.

Correct Answer: solid

Question 3:

The form y = a(x-h)^2 + k is known as ________ form.

Correct Answer: vertex

Question 4:

The solutions to a system of inequalities are found where the shaded regions ________.

Correct Answer: overlap

Question 5:

To determine which region to shade, choose a ________ point and test it in the inequality.

Correct Answer: test

Question 6:

The line that cuts the parabola in half is called the ________.

Correct Answer: axis of symmetry

Question 7:

The x-intercepts of a parabola are also called ________ or roots.

Correct Answer: zeros

Question 8:

The graph of a quadratic equation is a ________.

Correct Answer: parabola

Question 9:

The equation y = a(x-p)(x-q) is in ________ form.

Correct Answer: intercept

Question 10:

If the coefficient of the x^2 term is negative, the parabola opens ________.

Correct Answer: down

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.1.3:Graph a quadratic function. Identify the domain, range, x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology.

Teaching Materials

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