JOURNEY THROUGH QUADRATIC RELATIONS

By: Bansari Shah

What's Quadratics & How's it Useful

Ever been curious when you look at something that's curved? Have you ever

Basically the word quadratics comes from 'Quad' meaning square since the variables get squared ( ).

Table of Contents

Introduction

Key information about Parabola

  • Basic Information Facts About Parabolas

Intro to Parabolas

  • Key Features of Quadratic Relations

First & Second Differences


Types of equations

Vertex Form

Standard Form

Factored Form


Vertex Form

  • Key Facts
  • Transformations
  • Finding An Equation When Given the Vertex
  • Graphing from Vertex Form
  • Isolating for x

Standard Form

  • Solving Using Quadratic Equation
  • Graphing from Standard Form
  • Discriminant
  • Completing the square: vertex to standard form

Factored Form

  • Factors & zeros
  • Expanding - factored form to standard form
  • Common Factoring
  • Simple Trinomial
  • Complex Trinomial
  • Difference of Squares
  • Perfect Square
  • Factoring by grouping

Word Problems

  • Number Problems
  • Optimization Problems
  • Revenue Problems

Reflection

  • Unit Analysis


Helpful Links

Basic Important Facts About Parabolas

  • Parabolas can open either 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 the minimum or maximum 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.
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Key Features of Quadratic Relations

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First & Second Differences

We know that if it is a linear relation then the first differences will be the same but this time when the first differences aren't the same that means you have to find out the second differences. When the second differences are the same it's called a quadratic equation.
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Types of Equations

Vertex Form

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Standard Form

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Factored Form

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Vertex Form

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Key Facts

In the vertex form, the vertex (x,y) = (h,k)

The axis of symmetry : (x = h)

The optimal value is (y = k)

Note: When writing down the vertex, remember that the h value is always the opposite. If the h value is positive then when you write it down it would be negative and if the h value is negative then when you write it down it would be positive.

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Transformations

When in a vertex form each letter is responsible for a transformation.


'a' stretches the parabola vertical if this value is a whole number. If the value is a fraction than that would be a vertical compression. Furthermore, the 'a' value tells us whether the parabola will open up or down. If the "a" value is negative then the parabola will open down & if the 'a' value is positive then the parabola will open up.

(NOTE: if the 'a' value is negative, the vertex will be a maximum value & if the 'a' value is positive, the vertex will be a minimum value.)


'h' moves the vertex of the parabola either left or right. When 'h' is negative move right and when its positive move left.

(NOTE: it affects the horizontal translation)


'k' moves the vertex of the parabola up or down. If the 'k' value is negative the parabola opens down. (NOTE: affects the vertical translation. Also if the "a" value is negative there will be a reflection)

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Finding An Equation When Given Vertex

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Graphing From Vertex Form

Here are the following steps, which you need to follow in order to graph in vertex form.


  1. Find the vertex, since the formula is given in vertex form. (h,k)
  2. Find the y-intercept. In order to find the y-intercept you need to plug in the value of x to be 0.
  3. Find the x-intercept(s). To find the x-intercept, you need to plug in the value of y to be 0. You can solve for "x" by using the quadratic formula.
  4. Graph the parabola using the points which you found in steps 1- 3.
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Isolating For x

When you're isolating for x, you're basically finding the zeros or the x-intercepts in the vertex form. Here are the following which you need to follow in order to successfully do the method correctly.

Standard Form

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Solving Using Quadratic Equation

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Graphing From Standard Form

Discriminant

The discriminant is the equation which is inside the square root.

There are 3 possible solutions of a discriminant.


  1. When the discriminant is less than zero, there is no solution.
  2. When the value of discriminant is more than zero, there will be 2 solutions.
  3. When the discriminant is 0, there will be only one solution. Whether you add or subtract it from 'b', you will get the same answer.

Solution 1

When the discriminant is less than zero, there is absolute no solution.
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Solution 2

When the value of the discriminant is more than zero, there will be 2 solutions.
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Solution 3

When the discriminant is 0, there will be only one answer. Whether you add or subtract the 'b' you will get the same answer.
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Completing the Square : vertex to standard form

Factored Form

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Factors & Zeros

Expanding - factored form to standard form

In order for you to get from Factored Form to Standard Form, you would expand the terms in the first bracket to the second bracket. Once you're done expanding, you're going to see that you have like terms so all you would do is add those like terms up accordingly to the signs.
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Common Factoring

Simple Trinomial

Complex Trinomial

Difference of Squares

Perfect Squares

Factoring by Grouping

Word Problems

Number Problems

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Optimization Problems

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Revenue Problems

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Reflection

Throughout this unit, I learned quite a lot things such as what the three quadratic equations were. In relation, I learned how to go from one quadratic equation to another; for example from standard form to vertex form.
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Helpful Links

Quadratic Relations Grade 10 academic Lesson 4 1 4 2