Cracking the Code: Finding Quadratic Equations from Three Points

Algebra 2 Grades High School 3:36 Video
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Lesson Description

Learn how to determine the equation of a parabola in general form when given three points. This lesson uses systems of equations and elimination to solve for the coefficients a, b, and c.

Video Resource

Find the Equation of a Quadratic (Parabola) Given 3 Points

Mario's Math Tutoring

Duration: 3:36
Watch on YouTube

Key Concepts

  • General form of a quadratic equation (y = ax² + bx + c)
  • Systems of three linear equations
  • Solving systems of equations by elimination
  • Substitution

Learning Objectives

  • Students will be able to create a system of three linear equations from three given points on a parabola.
  • Students will be able to solve a system of three equations using elimination to find the coefficients a, b, and c.
  • Students will be able to write the equation of a quadratic function in general form given three points.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of a quadratic equation (y = ax² + bx + c). Briefly discuss how three points uniquely define a parabola. Introduce the idea that each point can be used to create an equation, leading to a system of equations.
  • Video Viewing & Note-Taking (10 mins)
    Play the Mario's Math Tutoring video: 'Find the Equation of a Quadratic (Parabola) Given 3 Points'. Instruct students to take notes on the steps involved in creating the system of equations and solving for a, b, and c. Encourage students to pause the video at key points to understand the calculations.
  • Guided Practice (15 mins)
    Work through an example problem similar to the one in the video, guiding students through each step. Emphasize the importance of accurate substitution and careful elimination. Address any questions or misconceptions that arise.
  • Independent Practice (15 mins)
    Provide students with 2-3 sets of three points and have them find the equation of the corresponding parabolas independently. Encourage them to work in pairs and check their answers with each other.
  • Wrap-up & Discussion (5 mins)
    Review the key steps and concepts covered in the lesson. Address any remaining questions. Preview upcoming topics related to quadratic functions.

Interactive Exercises

  • Online System Solver
    Use an online system of equations solver (e.g., Wolfram Alpha) to check the solutions obtained through elimination. This allows students to verify their work and identify any errors.
  • Graphing the Parabola
    Once the equation is found, have students graph the parabola using a graphing calculator or online tool (e.g., Desmos) and verify that the original three points lie on the graph.

Discussion Questions

  • Why do we need three points to define a parabola?
  • What are the advantages and disadvantages of using elimination to solve the system of equations?
  • Can you think of real-world scenarios where finding the equation of a parabola given points would be useful?

Skills Developed

  • Creating and solving systems of linear equations
  • Algebraic manipulation
  • Problem-solving
  • Critical Thinking
  • Attention to detail

Multiple Choice Questions

Question 1:

The general form of a quadratic equation is given by:

Correct Answer: y = ax² + bx + c

Question 2:

To find the equation of a parabola given three points, you need to solve a system of how many equations?

Correct Answer: 3

Question 3:

In the general form of a quadratic equation, which variable represents the y-coordinate?

Correct Answer: y

Question 4:

What method is primarily used in the video to solve the system of equations?

Correct Answer: Elimination

Question 5:

If you have solved for 'a', 'b', and 'c', what is the next step in finding the equation of the quadratic?

Correct Answer: Substitute a, b, and c back into the general form

Question 6:

What is the significance of the three points provided?

Correct Answer: They uniquely define the parabola

Question 7:

Which of the following is NOT a necessary step in finding the quadratic equation?

Correct Answer: Finding the derivative

Question 8:

When solving the system of equations, what operation is used to eliminate a variable?

Correct Answer: Both addition and subtraction

Question 9:

Given the equation y = ax² + bx + c, which coefficient affects the vertical stretch or compression of the parabola?

Correct Answer: a

Question 10:

Once you find the equation of the quadratic, how can you verify that it is correct?

Correct Answer: By plugging the original three points back into the equation

Fill in the Blank Questions

Question 1:

The equation y = ax² + bx + c is known as the _______ form of a quadratic equation.

Correct Answer: general

Question 2:

To solve for three variables (a, b, c), you need to create a system of _______ equations.

Correct Answer: three

Question 3:

The process of removing a variable from a system of equations by adding or subtracting equations is called _______.

Correct Answer: elimination

Question 4:

Once you find the values of a, b, and c, you _______ them back into the general form to get the equation.

Correct Answer: substitute

Question 5:

Each point (x, y) on the parabola provides one _______ to help solve for a, b, and c.

Correct Answer: equation

Question 6:

The values of a, b, and c in the quadratic equation determine the _______ and _______ of the parabola.

Correct Answer: shape/position

Question 7:

When multiplying an entire equation by a constant to prepare for elimination, it is important to _______ the constant to _______ term in the equation.

Correct Answer: distribute/every

Question 8:

After solving for two of the variables, the remaining variable can be found by _______ the known values into one of the equations.

Correct Answer: substituting

Question 9:

If a = 0 in the equation y = ax² + bx + c, the equation is no longer a _______, but a _______.

Correct Answer: quadratic/line

Question 10:

Checking your solution by plugging the original points back into the equation helps ensure you don't have any _______ errors.

Correct Answer: calculation

Educational Standards

A2.A.1.7:Represent and evaluate mathematical models using systems of linear equations with a maximum of three variables. Graphing calculators or other appropriate technology may be used. 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. 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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