Graphing Quadratic Inequalities in Vertex Form: A Visual Approach

Algebra 2 Grades High School 21:41 Video
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Lesson Description

Learn how to graph quadratic inequalities presented in vertex form, focusing on identifying the vertex, understanding inequality symbols, and shading the solution region.

Video Resource

Graphing Quadratic Inequalities Algebra | Vertex Form

Kevinmathscience

Duration: 21:41
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Key Concepts

  • Vertex form of a quadratic equation
  • Interpreting inequality symbols (solid vs. dashed lines)
  • Shading solution regions (above or below the parabola)
  • Symmetry of quadratic functions

Learning Objectives

  • Identify the vertex of a quadratic inequality given in vertex form.
  • Determine whether to use a solid or dashed line when graphing a quadratic inequality.
  • Determine whether to shade above or below the parabola based on the inequality symbol.
  • Graph a quadratic inequality in vertex form and shade the solution region.

Educator Instructions

  • Introduction (5 mins)
    Begin by briefly reviewing the standard form of a quadratic equation and how to graph it. Then, introduce the concept of quadratic inequalities and their graphical representation. Briefly introduce vertex form as an alternative.
  • Vertex Form Review (10 mins)
    Explain the vertex form of a quadratic equation: y = a(x - h)^2 + k, where (h, k) is the vertex. Emphasize how to easily identify the vertex from this form and the horizontal and vertical translations. Review how the 'a' value affects the parabola's direction and width.
  • Graphing Quadratic Inequalities (20 mins)
    Explain the steps to graph a quadratic inequality in vertex form: 1. Identify the vertex. 2. Create a table of values using symmetry. 3. Plot the points and draw the parabola (dashed or solid line based on the inequality). 4. Determine whether to shade above or below the parabola based on the inequality symbol (using the "crocodile method" or testing a point).
  • Examples (15 mins)
    Work through several examples of graphing quadratic inequalities in vertex form, demonstrating each step clearly. Include examples with different inequality symbols (>, <, ≥, ≤) and different vertex locations.
  • Practice (15 mins)
    Have students practice graphing quadratic inequalities in vertex form independently or in pairs. Provide feedback and answer questions.

Interactive Exercises

  • Vertex Identification Game
    Present students with several quadratic inequalities in vertex form and have them quickly identify the vertex. This can be done as a class activity or in small groups.
  • Shading Challenge
    Provide students with graphs of parabolas (both solid and dashed) and have them determine the correct shading based on given inequalities. They can use the crocodile method or test points to verify their answers.

Discussion Questions

  • How does the vertex form of a quadratic equation help us easily identify the vertex?
  • What is the significance of a solid versus a dashed line in the graph of a quadratic inequality?
  • How can you determine whether to shade above or below the parabola when graphing a quadratic inequality?
  • Can you explain why the table of values works based on the symmetry of the parabola?

Skills Developed

  • Graphing quadratic functions
  • Interpreting and applying inequalities
  • Using symmetry to efficiently graph functions
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the vertex of the quadratic inequality y > 2(x - 3)^2 + 1?

Correct Answer: (3, 1)

Question 2:

Which inequality symbol would require a dashed line when graphing?

Correct Answer: <

Question 3:

For the inequality y < (x + 2)^2 - 4, do you shade above or below the parabola?

Correct Answer: Below

Question 4:

What does the 'a' value in vertex form y = a(x - h)^2 + k affect?

Correct Answer: The direction and width of the parabola

Question 5:

Given the vertex (1, -2), what is the point symmetrical to (0, -1) across the axis of symmetry?

Correct Answer: (0, -3)

Question 6:

Which point would you choose to test if it is in the solution region of a quadratic inequality?

Correct Answer: (0, 0)

Question 7:

What does the value of 'h' in the vertex form y = a(x - h)^2 + k do to the graph?

Correct Answer: Horizontal Translation

Question 8:

What does the value of 'k' in the vertex form y = a(x - h)^2 + k do to the graph?

Correct Answer: Vertical Translation

Question 9:

You graph a quadratic inequality and use a dashed line. You shade the region above the parabola. What does this indicate?

Correct Answer: No points on the parabola are solutions, but all points above are.

Question 10:

You graph a quadratic inequality and use a solid line. You shade the region below the parabola. What does this indicate?

Correct Answer: All points on the parabola are solutions.

Fill in the Blank Questions

Question 1:

The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (____, ____) is the vertex.

Correct Answer: h, k

Question 2:

If the inequality symbol is ≥ or ≤, you use a ______ line when graphing.

Correct Answer: solid

Question 3:

If the inequality is y > f(x), you shade ______ the parabola.

Correct Answer: above

Question 4:

The axis of ______ divides the parabola into two symmetrical halves.

Correct Answer: symmetry

Question 5:

In the vertex form, the 'h' value represents a _______ shift.

Correct Answer: horizontal

Question 6:

A negative 'a' value in vertex form causes the parabola to open ____.

Correct Answer: down

Question 7:

In the crocodile method, the crocodile always tries to eat the ______ side.

Correct Answer: big

Question 8:

If the graph of a quadratic inequality has a vertex at (2, 3) and passes through (0, 0), the axis of symmetry is x = ____.

Correct Answer: 2

Question 9:

In vertex form, the 'k' value represents a _______ shift.

Correct Answer: vertical

Question 10:

For a quadratic function, there is _______ symmetry around the vertex.

Correct Answer: perfect

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.

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