Quadratic Relations

Final Submission || Harman Virk

Table of Contents


  1. The Parabola

Graphing Vertex Form:

  1. First and Second Differences
  2. Identifying Transformations of a Parabola
  3. Word Problem

Factored Form:

  1. Video of Factoring
  2. Different Types of Factoring and Examples
  3. Factoring to determine x-intercepts
  4. Word Problem

Standard Form:

  1. Completing the Square
  2. Quadratic Formula
  3. The Discriminant
  4. Word Problem


  1. Assessment
  2. Reflection

The Parabola

Everything you ever wanted to know about parabolas...

  • Parabolas can open up or down
  • The zero of a parabola is where the graph crosses the x-axis
  • "Zeros" can also be called "x-intercepts" or "roots"
  • The axis of symmetry divides the parabola into two equal halves
  • The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.
  • The optimal value is the value of the y co-ordinate of the vertex
  • The y-intercept of a parabola is where the graph crosses the y-axis

Graphing Vertex Form

First & Second Differences

First differences indicate that it's a linear equation. Second differences mean that it's quadratic. They must have equal second differences and unequal first differences.
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Identifying Transformations of a Parabola

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The 'a' value

If a > 0, the parabola opens up.

If a < 0, the parabola opens down.

If -1 < a < 1, the parabola is vertically compressed.

If a > 1 or a < -1 the parabola is vertically stretched.

The 'k' value

If k > 0, the vertex moves up by k units.

If k < 0, the vertex moves down by k units.

The 'h' value

If h > 0, the vertex moves to the right by h units.

If h < 0, the vertex moves to the left by h units.

Word problem

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Factored form

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Types of Factoring

Example 1: Factoring quadratics with a leading coefficient of 1 | Algebra II | Khan Academy

Different Types of Factoring & Examples

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  • The greatest common factor (GCF) is the largest integer that divides evenly into each of a given set of numbers.
  • When factoring polynomial expressions we must find the greatest common factor.
  • We look for the greatest common numerical factor, and for the variable with the highest degree of the variable common to each term.
  • We are looking for a whole number factor
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  • A simple trinomial is a quadratic where 'a' = 1
  • Given a quadratic in standard form, you can factor to get factored form
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  • 'a' does not equal 1
  • Always look at the common factor first when factoring trinomial
  • Not all quadratic expressions can be factored.
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  • Always look for a common factor first when factoring a trinomial
  • We have two different terms

Factoring to determine x-intercepts

1) Replacing y with 0 and factoring the quadratic

2) Setting each bracket = 0 and solving for x

y = x^2 + 12x +27

0 = (x + 9) (x + 3)

x + 9 = 0 --> x = -9

x + 3 = 0 --> x = -3

The x-intercepts are -3 and -9

Word Problem

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Standard Form

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Completing the Square

The process of completing the square involves changing the first two terms of a quadratic relation of the form y = ax^2 + bx + c into a perfect square while maintaining the balance of the original relation.

y = ax^2 + bx + c ----> y = a (x - h)^2 + k

3.14 Completing the square

Quadratic Formula

The quadratic makes it a lot easier to find the x-intercepts when in standard form
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The discriminant

When D > 0, there will be two solutions.

When D < 0, there will be zero solutions.

When D = 0, there will be one solution.

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Word Problem

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When we started to learn about the factored form, we took a quadratic relation in standard form and turned it into factored form. Both equations meant the same thing. In standard form and in factored form we can find the x-intercepts, but both are different and unique. In factored form, we can factor to find the x-intercepts while in standard form we must use the quadratic formula. All 3 give vital information for the parabolas in order to graph them. We learned how to go from one form to another.