Cracking Quadratics: Solving for Equations Given Three Points

Algebra 2 Grades High School 6:22 Video
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Lesson Description

Learn how to find the equation of a quadratic (parabola) when given three points by solving a system of three equations with three variables. This lesson reinforces systems of equations and quadratic functions.

Video Resource

Find Equation of Quadratic (Parabola) Given 3 Points (System of 3 Equations)

Mario's Math Tutoring

Duration: 6:22
Watch on YouTube

Key Concepts

  • Quadratic Functions: Understanding the standard form y = ax² + bx + c.
  • Systems of Equations: Solving systems of three equations with three variables.
  • Elimination Method: Using elimination to solve systems of equations.

Learning Objectives

  • Students will be able to set up a system of three equations from three given points on a parabola.
  • Students will be able to solve a system of three equations using the elimination method.
  • Students will be able to determine the quadratic equation in the form y = ax² + bx + c, given three points.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a quadratic equation (y = ax² + bx + c) and emphasize that the goal is to find the values of a, b, and c. Briefly discuss how three points uniquely define a parabola. Show a graph of a parabola and highlight how each point satisfies the equation.
  • Video Demonstration (15 mins)
    Play the YouTube video 'Find Equation of Quadratic (Parabola) Given 3 Points (System of 3 Equations)' by Mario's Math Tutoring. Pause at key points to explain each step of the process, especially the setup of the system of equations and the elimination method. Encourage students to take notes.
  • Guided Practice (20 mins)
    Work through an example problem together as a class. Provide three points and guide students through the steps of setting up the system of equations, eliminating variables, and solving for a, b, and c. Emphasize careful arithmetic and organization. Have students come to the board to help solve.
  • Independent Practice (15 mins)
    Provide students with a set of three points and have them independently solve for the quadratic equation. Circulate to provide assistance and answer questions. Have students compare their answers with a partner to check for accuracy.
  • Wrap-up and Assessment (5 mins)
    Review the key steps and concepts. Announce the multiple choice and fill in the blank quiz.

Interactive Exercises

  • Point-Parabola Matching
    Present students with a set of parabolas and a set of points. Students must match three points to the correct parabola.
  • Equation Builder
    Provide students with the values of 'a', 'b', and 'c', and have them construct the quadratic equation. Then, provide points and have them check whether the points lie on the parabola represented by the equation.

Discussion Questions

  • Why do we need three points to define a quadratic equation?
  • What are the advantages and disadvantages of using the elimination method for solving systems of equations?
  • How does the value of 'a' in the quadratic equation affect the shape of the parabola?

Skills Developed

  • Problem-solving: Applying algebraic techniques to solve real-world problems.
  • Analytical Thinking: Analyzing and interpreting mathematical information.
  • System Thinking: Connecting systems of equations and quadratic relationships.

Multiple Choice Questions

Question 1:

The standard form of a quadratic equation is:

Correct Answer: y = ax² + bx + c

Question 2:

How many points are needed to uniquely define a parabola?

Correct Answer: 3

Question 3:

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

Correct Answer: Elimination

Question 4:

In the quadratic equation y = ax² + bx + c, what does 'a' represent?

Correct Answer: Coefficient affecting the parabola's width and direction

Question 5:

When solving a system of three equations, you eliminate one variable to create:

Correct Answer: Two equations with two variables

Question 6:

If you multiply an equation by -1, what happens to the signs of the terms?

Correct Answer: They all change to the opposite sign

Question 7:

After solving for 'a' in the system, what is the next logical step?

Correct Answer: Substitute 'a' back into an equation with two variables to solve for another variable

Question 8:

What is the purpose of eliminating variables in a system of equations?

Correct Answer: To reduce the number of variables and equations until you can solve for one variable

Question 9:

What happens to the C values in the video when using the elimination method?

Correct Answer: They are added together

Question 10:

Which step do you use to check your values of a, b, and c?

Correct Answer: Plug the values back into the original equations to confirm that the points work

Fill in the Blank Questions

Question 1:

The goal is to find the values of ___, ___, and ___ in the quadratic equation.

Correct Answer: a, b, c

Question 2:

The graph of a quadratic equation is called a ________.

Correct Answer: parabola

Question 3:

The ________ method involves adding or subtracting equations to eliminate variables.

Correct Answer: elimination

Question 4:

Substituting values back into the original equations is a way to ________ your solution.

Correct Answer: check

Question 5:

First, you create three equations by using the y values and plugging the x values into the equation y = ax² + bx + c. You will _______ a for x and y.

Correct Answer: substitute

Question 6:

When multiplying an equation by a constant, you must distribute that constant to ______ term in the equation.

Correct Answer: every

Question 7:

If a variable cancels out during the elimination process, it means that term is effectively being set to _______ .

Correct Answer: zero

Question 8:

The points need to be non-________ to be able to define a single equation.

Correct Answer: collinear

Question 9:

In the equation y = ax² + bx + c, the 'c' term represents the ____-intercept of the parabola.

Correct Answer: y

Question 10:

Once you find the values of a, b, and c, you _______ them back into the standard quadratic equation to get the final answer.

Correct Answer: substitute

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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