Unlocking the Secrets of Parabolas: Writing Quadratic Equations from Vertex and a Point

Algebra 2 Grades High School 2:28 Video
Print Lesson Plan Watch Video

Lesson Description

Learn how to determine the equation of a parabola in vertex form when given its vertex and one additional point. This lesson will guide you through the process step-by-step, reinforcing key concepts of quadratic functions.

Video Resource

Write the Quadratic Equation for the Parabola Given the Vertex and a Point

Mario's Math Tutoring

Duration: 2:28
Watch on YouTube

Key Concepts

  • Vertex form of a quadratic equation: y = a(x - h)² + k
  • Vertex of a parabola: (h, k)
  • Using a point on the parabola to solve for the 'a' value

Learning Objectives

  • Students will be able to identify the vertex (h, k) from a given parabola or description.
  • Students will be able to substitute the vertex and a given point into the vertex form of a quadratic equation.
  • Students will be able to solve for the 'a' value in the vertex form equation.
  • Students will be able to write the quadratic equation in vertex form given the vertex and another point on the parabola.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of a quadratic equation and its graph (parabola). Introduce the vertex form: y = a(x - h)² + k, explaining that (h, k) represents the vertex. Briefly discuss the significance of the 'a' value.
  • Video Explanation (10 mins)
    Play the YouTube video 'Write the Quadratic Equation for the Parabola Given the Vertex and a Point' by Mario's Math Tutoring. Encourage students to take notes on the key steps.
  • Worked Example (10 mins)
    Work through the example from the video on the board, emphasizing each step: identifying the vertex, substituting the vertex and point into the vertex form, solving for 'a', and writing the final equation. Explain the logic behind each step.
  • Guided Practice (10 mins)
    Provide students with a similar problem: Vertex (1, 2), Point (3, 6). Guide them through the steps, asking questions and providing hints as needed. Encourage students to work in pairs.
  • Independent Practice (10 mins)
    Give students another problem to solve independently: Vertex (-2, -1), Point (0, 3). This will allow you to assess individual understanding.
  • Wrap-up and Q&A (5 mins)
    Review the key concepts and answer any remaining questions. Briefly mention the connection to standard form and how to convert between the two (as mentioned in the video).

Interactive Exercises

  • Graphing Parabolas Activity
    Using a graphing calculator or online tool (like Desmos), have students graph different parabolas in vertex form. They should change the values of 'a', 'h', and 'k' and observe how the graph changes. This will visually reinforce the effect of each parameter.
  • Error Analysis
    Present a problem where the 'a' value was incorrectly calculated. Have students find the error and explain the correct solution. This promotes critical thinking.

Discussion Questions

  • Why is the vertex form useful for representing parabolas?
  • How does the 'a' value affect the shape of the parabola?
  • What happens if you are given the vertex and *two* other points on the parabola? Could you still find the equation? Why or why not?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Analytical thinking
  • Understanding of quadratic functions

Multiple Choice Questions

Question 1:

The vertex form of a quadratic equation is given by:

Correct Answer: y = a(x - h)² + k

Question 2:

In the vertex form y = a(x - h)² + k, what does (h, k) represent?

Correct Answer: The vertex

Question 3:

What is the first step in finding the equation of a parabola in vertex form when given the vertex and a point?

Correct Answer: Substitute the vertex into the vertex form

Question 4:

If the vertex of a parabola is (2, -3), what values do you substitute for h and k in the vertex form?

Correct Answer: h = 2, k = -3

Question 5:

After substituting the vertex and the given point into the vertex form, what are you solving for?

Correct Answer: The 'a' value

Question 6:

A parabola has a vertex at (0, 4) and passes through the point (2, 0). What is the value of 'a'?

Correct Answer: -1

Question 7:

Given a vertex of (-1, 5) and a point (1, 9), which equation represents the parabola in vertex form?

Correct Answer: y = (x + 1)² + 5

Question 8:

The vertex form of a quadratic equation helps easily identify which feature of the parabola?

Correct Answer: The vertex

Question 9:

What does the 'a' value in the vertex form indicate about the parabola?

Correct Answer: The direction and width

Question 10:

Why is it important to use a point other than the vertex to determine the equation of the parabola?

Correct Answer: To determine the 'a' value

Fill in the Blank Questions

Question 1:

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

Correct Answer: vertex

Question 2:

To find the equation of a parabola in vertex form, you need the ________ and another point on the parabola.

Correct Answer: vertex

Question 3:

In the equation y = a(x - h)² + k, the value of 'a' determines the direction and ________ of the parabola.

Correct Answer: width

Question 4:

If the vertex is (-3, 1), then h = _______ and k = _______.

Correct Answer: -3, 1

Question 5:

After substituting the vertex and a point into the vertex form, you solve for the variable _______.

Correct Answer: a

Question 6:

A parabola with a vertex at (5, -2) passes through the point (6, -1). The 'a' value is _______.

Correct Answer: 1

Question 7:

The vertex form is useful because it directly shows the _______ of the parabola.

Correct Answer: vertex

Question 8:

If 'a' is negative, the parabola opens _______.

Correct Answer: downward

Question 9:

Substituting a point and the vertex allows us to find the specific _______ of the parabola.

Correct Answer: equation

Question 10:

The axis of symmetry of the parabola passes through the _______.

Correct Answer: vertex

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.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans