Mastering Parabolas: Graphing Quadratic Functions in Vertex Form

Algebra 2 Grades High School 7:47 Video
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Lesson Description

Learn to graph parabolas in vertex form by identifying the vertex, axis of symmetry, and understanding transformations. This lesson covers key features like maximum/minimum values, domain, range, and intervals of increase/decrease.

Video Resource

Graphing Parabolas in Vertex Form (3 Examples)

Mario's Math Tutoring

Duration: 7:47
Watch on YouTube

Key Concepts

  • Vertex Form of a Quadratic Function: y = a(x - h)^2 + k
  • Vertex: (h, k)
  • Axis of Symmetry: x = h
  • Transformations: Horizontal and vertical shifts, vertical stretch/compression, reflection across the x-axis
  • Domain and Range
  • Maximum or Minimum Value
  • Intervals of Increasing and Decreasing

Learning Objectives

  • Students will be able to identify the vertex, axis of symmetry, and transformations from the vertex form of a quadratic equation.
  • Students will be able to graph parabolas accurately using the vertex and additional points.
  • Students will be able to determine the domain, range, maximum or minimum value, and intervals of increasing and decreasing for a given parabola.
  • Students will be able to find the y-intercept of a parabola in vertex form.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of a quadratic equation and its graph (a parabola). Introduce the vertex form: y = a(x - h)^2 + k. Explain that this form makes it easy to identify the vertex and transformations. Briefly explain what will be covered in the lesson.
  • Video Presentation (15 mins)
    Play the Mario's Math Tutoring video 'Graphing Parabolas in Vertex Form (3 Examples)'. Instruct students to take notes on the key concepts and examples presented. Emphasize the importance of understanding how 'h', 'k', and 'a' affect the graph.
  • Guided Practice (20 mins)
    Work through the examples from the video again, but this time pause after each step and ask students guiding questions. Have them actively participate in determining the vertex, axis of symmetry, and plotting points. Reinforce the connection between the equation and the graph.
  • Independent Practice (15 mins)
    Provide students with a set of practice problems where they need to graph parabolas in vertex form and identify key features. Circulate the classroom to provide assistance as needed.
  • Wrap-up and Assessment (5 mins)
    Review the main concepts of the lesson. Answer any remaining questions. Administer the multiple-choice or fill-in-the-blank quiz to assess student understanding.

Interactive Exercises

  • Vertex Form Exploration
    Use a graphing calculator or online graphing tool (like Desmos) to explore how changing the values of 'a', 'h', and 'k' in the vertex form affects the graph of the parabola. Students can experiment with different values and observe the resulting transformations.
  • Matching Game
    Create a matching game where students have to match vertex form equations with their corresponding graphs or key features (vertex, axis of symmetry).

Discussion Questions

  • How does changing the value of 'a' in the vertex form affect the parabola?
  • How does the vertex form make it easier to graph a parabola compared to the standard form?
  • Why is the axis of symmetry important for graphing parabolas?
  • Can a parabola have two y-intercepts? Why or why not?

Skills Developed

  • Graphing quadratic functions
  • Identifying key features of parabolas
  • Applying transformations to functions
  • Analytical thinking
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the vertex of the parabola y = 2(x - 3)^2 + 1?

Correct Answer: (3, 1)

Question 2:

What is the equation of the axis of symmetry for the parabola y = -(x + 2)^2 - 4?

Correct Answer: x = -2

Question 3:

If 'a' is negative in the vertex form y = a(x - h)^2 + k, the parabola opens:

Correct Answer: Downward

Question 4:

Which of the following transformations does 'h' in y = a(x - h)^2 + k represent?

Correct Answer: Horizontal shift

Question 5:

What is the range of the parabola y = (x - 1)^2 + 3?

Correct Answer: y ≥ 3

Question 6:

What is the domain of the parabola y = -3(x + 4)^2 - 2?

Correct Answer: All real numbers

Question 7:

Which equation has a minimum value of 5?

Correct Answer: y = (x-2)^2 + 5

Question 8:

In the equation y = a(x - h)^2 + k, what does 'k' represent?

Correct Answer: The y-coordinate of the vertex

Question 9:

Which parabola is wider than y = x^2?

Correct Answer: y = 0.5x^2

Question 10:

Which parabola opens downward and has a vertex at (0,3)?

Correct Answer: y = -(x-0)^2 + 3

Fill in the Blank Questions

Question 1:

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

Correct Answer: h

Question 2:

The axis of symmetry is a vertical line that passes through the _______ of the parabola.

Correct Answer: vertex

Question 3:

If 'a' is positive, the parabola opens ________.

Correct Answer: upward

Question 4:

The _________ of a function are the set of all possible input values (x-values).

Correct Answer: domain

Question 5:

The _________ of a function are the set of all possible output values (y-values).

Correct Answer: range

Question 6:

When the parabola has a highest point, it is called a ________.

Correct Answer: maximum

Question 7:

When the parabola has a lowest point, it is called a ________.

Correct Answer: minimum

Question 8:

The y-coordinate of the vertex represents the maximum or minimum _________ of the function.

Correct Answer: value

Question 9:

A negative 'a' value reflects the parabola across the _______ axis.

Correct Answer: x

Question 10:

The point where the parabola intersects the y-axis is called the _______.

Correct Answer: y-intercept

Educational Standards

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. 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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