Quadratic Relations By: Ikhlas
Flyer for grade 10 math students
What is Quadratics and how is it useful?
The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).
It is also called an "Equation of degree 2" (because of the "2" on the x). Quadratic are visually represented in a graph by U shaped lines called Parabolas. A Parabolas is a curve where any point has an equal distance (except for the optimal value). Thinking about it, you may be asking yourself, "how is this useful?". Quadratics is very useful in all levels of study from business to science. If you want to find the maximum revenue a company can make, or to see how high you can throw a football before it lands, you would use quadratics.
Table of Content
Part 1
difference between linear and quadratics
Equations/formulas
vertex form
factor form
optimal values
Part 2
Formulas
Multiplying Polynomials
Factoring
Factoring by grouping
Factoring simple trinomials
Factoring Complex trinomials
Perfect squares
difference of squares
Part 3
Formulas
Completing the square
Solving Quadratic Equations
Finding points
The Quadratic formula
Discriminants
Word problems
Economic problem
Area and perimeter problem
motion problem
Maximize the Area of a fence with the given perimeter
Reflection
Terms and definitions
Parts of a Parabola Axis of symmetry: A line that vertically divides the graph from the middle. Zeros: points that land directly on the x-axis, or y-axis. Example; (3,0). Optimal value: the maximum or minimum value on the parabola. | describing properties of a parabola Horizontal Shift: When the parabola moves to the negative or to the positive (left or right) on the x-axis. Vertical Shift: When the parabola moves to the negative or positive on the y-axis (up or down). Stretch Compress Direction of opening | Terms for quadratic equations Factoring y=a(x-h)^2=k a:determines if the graph will vertically stretch or compress h:determines how many units to horizontally move the parabola k:determines how many unites to move on either the y or x axis. x-intercepts:the value of x when y=0 |
Parts of a Parabola
Zeros: points that land directly on the x-axis, or y-axis. Example; (3,0).
Optimal value: the maximum or minimum value on the parabola.
describing properties of a parabola
Vertical Shift: When the parabola moves to the negative or positive on the y-axis (up or down).
Stretch
Compress
Direction of opening
Part 1
Formulas
factor form: y=a(x-r)(x-s)
determining the equation given the Vertex
Vertex form
Value "a" value "a" determines weather the graph will stretch or compress. If the Value is a whole number, then it will stretch, if it is a fraction or a decimal, then it will compress. | Value "k" Value "k" determines weather the graph will move up, or down. If the value is negative, then the graph will move down, if positive, it will move up. | Value "h" Value "h" determines the horizontal shift of the graph. If the value is positive, then the graph will move to the right, if it's negative, it will move to the left. |
Value "a"
Value "k"
factor form
Optimal values
Finding optimal Values When given a graph, often times you are asked to find the optimal value. Weather the graph is inverted or not, the optimal value is the highest, or lowest value on the graph. The optimal value and the vertex are the same thing. On this graph, the optimal value is (1,-4) |
Part 2
solving Formulas
a^2-2ab+b^2-->(a-b)^2
(a^2-b^2)-->(a+b)(a-b)
Multiplying Polynomial
- To multiply polynomials, you need to multiply each term in one polynomial by each term in the other polynomial. Add those answers together, and simplify if needed. Often times you are given bigger equations that needs to be expanded and simplified, for example: 2(3x+5)(2x-6)-6(3x-6)(4x+5). Although it may look tantalizing, the same steps are required to solve these bigger equations (also make sure to use BEDMAS).
Factoring polynomials
factoring simple trinomials
factoring complex trinomials
perfect squares
difference of squares
Part 3
Completing the square
Solving quadratic equations
Factor Form
When solving a quadratic equation, your goal is to find the x-intercepts of the equation. When given an equation that is in factor form, it is very easy to solve. Example: (x+3)(x+4)=0 (remember to find the x-intercepts, you have to set the y value to zero). All you have to do is rearange the equation so that x is isolated. x=-3 & x=-4Vertex
If you want to be able to find the vertex of 2x^2+10+14 what you have to do is turn the equation into standard form by completing the square. After completing the square, your equation should look like this:2(x+5)^2-39. You can find the vertex by looking at the h,k values. The vertex for this equation is -5 and -39.If you want to find the aos, you can use the formula -b/2a by just plugging in the numbers and solving.With the aos, you can find the optimal value just by plugging in the aos value (x) into the formula and solving.
The Quadratic Formula
You might be confused by the +and- sign next to the "b" value. What you do is add, and subtract the product so that you have to answers (2 x-intercepts).
Discriminants
Word problems
Optimization Problem
Area and Perimeter problem
Motion problem
a) How far under water is the cannon?
b)how long will the cannon be above water?
c) what is the maximum height the cannon ball will travel.
Maximize the Area of a fence with the given perimeter
Reflection
Test Reflection
This was the Quadratics Mini test 1. I performed poorly on this test but I am aware of my mistakes and have improved. This test mainly covered everything in Part 1 of this flyer.Unit reflection
Quadratics was a very useful unit, and it is very obvious how it can applied in a variety of real-life application. It can be used for maximizing the revenue of a business, maximizing the area of a fence with a given perimeter, and is great for many other applications as well. The Quadratic formula was also a very useful and important formula when trying to find the x-intercepts. The basics of grade 10 quadratics will be pursued in the further years of our academic lives, so it is very important that we understand the basics now. If you need help with understanding quadratics, here is a links that might be of some assistance