All About Quadratics

Everything You Need To Know About Grade 10 Quadratics!

Table Of Contents


Introduction
- What is a Quadratic? When is it useful?
- Quadratics vs Linear Relationships (First and Second Differences)
- Base function (y= x²)
Vertex form
- Identifying transformations from the equation.
- Writing an equation, given transformations and/or graph.
- Graphing transformations of quadratic relations:

  • Mapping Formula
  • Step Notation
- Finding the y and x intercepts (zeros) in Vertex Form
Factored form
1. Expanding (Factored Form to Standard Form)
- Multiplying Binomials (FOIL)
- Special Products:
  • Perfect Square
  • Difference of Squares



2. Factoring (Standard Form to Factored Form)
- Common Factors

  • Monomial Common Factors (GCF)
  • Binomial Common Factors
  • Factor by Grouping
- Simple Factoring (a=1)
- Decomposition (a≠1) ​
3. Graphing Factored Form​
Standard form
- Completing the Square (Standard to Vertex Form)
- Quadratic Formula (To Solve)
- Discriminant (Number of Solutions)
- Graphing using the standard form
Word problems
- Economic Problems (Profit, Revenue...)
- Measurement Problems (Rectangles, Triangles...)
- Number Problems (Integers,
- Ball/Rocket/Fireworks Problems
- Optimization Problems

What is a Quadratic and When is it useful?

Quadratic comes from "quad" meaning square. This is because the variable gets squared (example: x²).
A Quadratic relation is the opposite of a Linear relation, because the shape of the graph is always a curve, whereas in linear it is a straight line.
Quadratics are graphed when creating graphs for profit or a function.
Everyday quadratics include the flight of a ball or a firework, or a graph of total profit made in businesses.

The shape made by the Quadratic function is also known as a Parabola.
A parabola has many different components:
  • Vertex (x,y) -- the point where the parabola changes direction.
  • The Optimal Value (OV)-- the y-value of the vertex.
  • The Axis of Symmetry (AOS)-- the x-value of the vertex, and passes through the x-value.
  • Zero--is/are the X-Intercept(s) of the parabola.
  • Y-Intercepts-- is the point where the parabola touched the y-axis.
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Similarities in Quadratic and Linear Functions


Linear Relation:


  • Y=mx+b
  • M represents the slope, steepness and direction.
  • B represents the y intercept and x=0 y=b.
  • How can you tell whether an equation is a quadratic relation or a linear relation, just by looking at it?
    By looking at a table of values, you can easily see whether an equation is a quadratic relation or a linear relation. This is because, by doing first and second difference, you can determine the type of relation. If the first difference is constant (Image 1), then it is a linear relation. If the first difference isn't constant, but the second difference is (Image 2), then it is a quadratic relation.
  • Standard Form is the original way to write the quadratic function. The equation could tell you the shape of the graph, the direction, and the y-intercept.
    Written as: ax²+bx+c
    Vertex Form is one of the ways to write a quadratic function. Just by looking at the equation, you could identify the transformations of the base graph, the vertex, the direction of opening...
    Written as: y= a(x-h)²+k ​
    Factored Form is simplified version of the standard equation. By looking at the equation, you can see the shape and direction of the graph, and the x-intercepts or zeros.
    Written as: y=a(x-r)(x-s)

Quadratic Relation:

y=ax^2+bx+c

  • A= slope
  • c=y intercept

Identifying Transformations from the Equation ​


The vertex form allows you to see the different transformations that are applied to the base parabola of y=x². These transformations could be identified from the equation, when it is in Vertex Form.

Transformations, from y= a(x-h)²+k ​:

The value (-h) that is inside the bracket, is the horizontal translation. The value of h, moves the parabola either left or right depending or it's sign. (If the sign is negative, it moves right, if it is positive move left)--Opposite operation.

The value of (k), is the vertical translation. It moves the parabola either up or down depending on the sign. (If the sign is negative, it moves down and if it is positive, it moves up).

The value of (a), is either a vertical stretch or compression, by the factor of (a). It would be a stretch, if the number was a whole number. If the value was either a fraction or a decimal, then it would be a compression. If the value is negative, then it means that the parabola was reflected over the X-axis.

Finding the Zeros and Y-Intercept

To solve a quadratic equation, means to find the x-intercept or the roots. There could be many possible solutions, however there would only be one y-intercept.


To find the X-Intercept(s) or the Y-Intercept, set the opposite variable to 0.

Graphing the Vertex Form

To graph a parabola that is in vertex form, you need to know the translations (of a, h and k), which is found in the equation. There are two methods on how to graph the vertex form; Mapping Notation and the Step Pattern.
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Learning Goals

I am able to...

  • identify the intercepts
  • differences and similarities between a quadratic relation.
  • identify the different types of form

Graphing Quadratics Using Step Patterns

Factored Form

Parts of the ​y=a(x-r)(x-s) ​ form

- The value of ​a gives you the shape and direction of opening
- The value of ​​r ​and s ​gives the two x-intercepts (r,0) and (s,0)
- To find the y-intercepts (0,y), set x as 0, and solve for y.
- ​
To find the vertex, use the zeros to find the AOS, and sub this value of x into the given equation to solve for y. ​​​​​​

Factoring Polynomials

Methods of factoring.
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Expanding Polynomials

1. Multiplying Binomials -- Use the distributive property to multiply binomials.​
2. Special Cases:
Perfect Squares -- when a binomial is squared.
Rule: First term is squared, plus 2 times the product of the first and second term, plus second term squared. (NOTE: or subtract if number is negative.)
Difference of Squares-- when everything in both brackets is the same, except one is being added, and the other subtracted.
Rule: First term squared, minus second term squared.
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Graphing Factored Form

Steps to Graph:
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To see an example on how to graph factored form, click the link below.
Graphing Parabolas in Factored Form y=a(x-r)(x-s)

Word Problem:

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Learning goals

I am able to....

use one out of the two methods for factoring

identify the equation within an word problem

Standard Form

Standard forms is always in the form ax

2

+ bx + c = 0 ."a","b" and "c" are the known values "a" cannot be zero no matter what. "x" is the variable that is unknown.

Quadratic Formula

All quadratic equations of the form ax²+bx+c,can be solved using the quadratic formula. It is used by substituting the values of a, b and c from the original equation, into the formula, then solving. ​
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Completing the Square

We can ...
Standard Form ---------- Vertex Form
ax²+bx+c ---------- y= a(x-h)²+k
... by completing the square.

Below is a video that can help you explain the steps in completing squares,

❖ Completing the Square - Solving Quadratic Equations ❖

Discriminant

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The discriminant is everything under the square root, by solving only this, you can know how many x-intercepts the equation will have.

Steps of the Quadratic Formula

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Graphing Standard Form

Steps:
  1. Find the x-intercepts of the equation, by using the quadratic formula.
  2. Find the vertex of the equation, by completing the square.
  3. Graph the x-intercepts and the vertex, then draw the parabola.
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Word Problems

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Learning goals

I am able to...

to solve an equation

identify the h, x, a and k in the equations

Reflection

Overall, the quadratics unit was fairly easy.
Even though there were many points where I just wanted to quit. However, the only thing that kept me going was that I would need Grade 10 Math in order to take Grade 11 University Math
The Vertex form unit was really just the beginning, but it was also one of the easiest units. The concepts were easy to understand, and I put a lot of effort into understanding the concepts of the vertex form. Factored form was what scared me, I had heard a lot of things about the unit, like that not many people really liked it and that it was the "hardest" of the three units. At first, this was really true, I found it hard to distinguish between the many different ways to factor. At some point I found the factoring unit easy and some part confusing. The unit I hated the most was the Standard form unit, and this was because I was in the mindset that it was easy, and therefore was behind on the homework. Afterwards, I crammed all the homework on one day, and I ended up doing somewhat better throughout the unit.


Therefore in all, the Quadratics unit went pretty good. The unit test on Vertex Form was my best test, and I am really proud of it


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Quadratic Formula