Cracking the Code: Finding Quadratic Equations from Three Points

Algebra 2 Grades High School 5:56 Video
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Lesson Description

Learn how to determine the equation of a parabola (quadratic) given three points by solving a system of three equations with three variables. This lesson reinforces algebraic manipulation and problem-solving skills.

Video Resource

Find the Equation for a Quadratic (Parabola) Given 3 Points (Not the Vertex)

Mario's Math Tutoring

Duration: 5:56
Watch on YouTube

Key Concepts

  • Quadratic Equations (Parabolas)
  • Systems of Equations (3x3)
  • Elimination Method
  • Substitution Method

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 3x3 system of equations using the elimination method.
  • Students will be able to determine the quadratic equation in the form y = ax^2 + bx + c given three points.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the standard form of a quadratic equation (y = ax^2 + bx + c) and the general shape of a parabola. Introduce the problem: finding the equation given three points instead of the vertex.
  • Video Viewing (15 mins)
    Play the Mario's Math Tutoring video: 'Find the Equation for a Quadratic (Parabola) Given 3 Points (Not the Vertex)'. Encourage students to take notes on the steps involved.
  • Step-by-Step Walkthrough (15 mins)
    Reiterate the steps from the video, writing them on the board. Explain how each point (x, y) is substituted into y = ax^2 + bx + c to create an equation. Emphasize simplifying the equations. Demonstrate the elimination method, highlighting strategic variable selection for elimination. Show back substitution after solving for one variable.
  • Alternative Method: Substitution (10 mins)
    Briefly explain how to solve this system using substitution. It is a good exercise to compare and contrast elimination versus substitution and ask students to express preferences for each method.
  • Guided Practice (15 mins)
    Work through a new example problem together as a class. Involve students by asking them to suggest which variable to eliminate and guiding them through the arithmetic. Have students come up to the board to solve steps.
  • Independent Practice (15 mins)
    Assign a similar problem for students to solve individually. Circulate to provide assistance. Allow them to work in pairs to compare answers.
  • Wrap-up (5 mins)
    Review the key steps and address any remaining questions. Preview the connection to real-world applications of quadratic equations.

Interactive Exercises

  • Error Analysis
    Present a worked-out problem with a mistake in the elimination or substitution process. Ask students to identify and correct the error.
  • Online System Solver
    Have students use an online system of equations solver (after they attempt the problem manually) to check their answers and understand how technology can assist in problem-solving.

Discussion Questions

  • Why do we need three points to define a unique parabola?
  • What are the advantages and disadvantages of using the elimination method vs. other methods (like substitution) for solving systems of equations?
  • Can you think of any real-world scenarios where you might need to find the equation of a parabola given some data points?

Skills Developed

  • Algebraic Manipulation
  • Problem-Solving
  • Systematic Thinking
  • Critical Thinking

Multiple Choice Questions

Question 1:

The general form of a quadratic equation is:

Correct Answer: y = ax^2 + bx + c

Question 2:

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

Correct Answer: 3

Question 3:

In the equation y = ax^2 + bx + c, what does 'a' represent?

Correct Answer: A coefficient that affects the parabola's shape and direction

Question 4:

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

Correct Answer: Elimination

Question 5:

If you substitute the point (1, 5) into the equation y = ax^2 + bx + c, what equation do you get?

Correct Answer: a + b + c = 5

Question 6:

What is the primary goal of the elimination method?

Correct Answer: To eliminate one variable at a time

Question 7:

After solving for 'a', 'b', and 'c', what do you do with those values?

Correct Answer: Plug them back into y = ax^2 + bx + c

Question 8:

What is the first step in setting up the system of equations?

Correct Answer: Substitute each point into the quadratic equation

Question 9:

What is meant by 'back substitution'?

Correct Answer: Substituting already-solved variables to find other variables

Question 10:

If a=2, b=-1, and c=3, what is the quadratic equation?

Correct Answer: y = 2x^2 - 1x + 3

Fill in the Blank Questions

Question 1:

The standard form of a quadratic equation is y = _______ + bx + c.

Correct Answer: ax^2

Question 2:

To find the equation of a quadratic, you need to substitute the x and y values of each point into the general equation to create a system of _______.

Correct Answer: equations

Question 3:

The _________ method involves adding or subtracting equations to eliminate one variable at a time.

Correct Answer: elimination

Question 4:

After solving for one variable, you use _______ to find the values of the other variables.

Correct Answer: back substitution

Question 5:

A parabola is the graphical representation of a _______ equation.

Correct Answer: quadratic

Question 6:

If you have the point (2,4) then x = ____ and y = ____.

Correct Answer: 2, 4

Question 7:

The constants 'a', 'b', and 'c' determine a parabola's _________ and ________.

Correct Answer: shape, location

Question 8:

A system of three equations with three variables requires __________ to solve it.

Correct Answer: three equations

Question 9:

When multiplying to eliminate a variable, remember to multiply _______ sides of the equation.

Correct Answer: both

Question 10:

If you eliminate a variable, the number of equations will be ___________ than the number of variables.

Correct Answer: greater

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.

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