Quadratics Relations

By: Simran Heer

What is Quadratics?

The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and differ in width or steepness, but they all have the same basic "U" shape. Quadratics can be used to graph many different types of data.

Table of Contents

In grade 9 we had learnt about graphing and analyzing linear relation, and now we have moved on to many components of quadratics, such as the basics of graphing these equations in variety of ways.


Introduction:


  • Second Differences
  • Properties of Parabolas
  • Transformations In Vertex Form


Types of Equations:


  • Factored Form: a(x-r)(x-s)
  • Standard Form: ax² + bx + c
  • Vertex Form: a(x-h)² + k


Word Problems:


  • 2 examples

Reflection:


  • Test

Second Differences

Finding the second differences indicates whether the equation is quadratic or not. A quadratic equation will always have equal second differences, and unequal first differences. When comparing this to a linear equation you will notice that, a linear equation always has equal first differences.
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Properties of Parabolas

A parabola is created, when a quadratic equation is graphed. Parabolas have:



  • Vertex: where the graph changes direction (x, y)
  • Axis of Symmetry: the line which passes when x=0, and divides the parabola into two equal parts
  • Optimal Value: the y coordinate, either the highest (max) or lowest (min) point
  • X-intercepts: the points which cross the x axis. There could be 2 solutions, 1 solution, or no solution.
  • Y-intercepts: when the graph crosses the y axis

Types of Equations

Factored Form: a(x-r)(x-s)

When using these equations, you deal with factoring and solving for the zeros. A zero of a parabola is another way to say x-intercept, and in order to find the solutions, you need to set y=0.


To graph a parabola in factored form, you need to:
Find the Axis Of Symmetry:

  • Find the zeros
  • Solve by adding both zeros, then dividing by 2. This will give you the AOS

How to find the Vertex:

  • Substitute the value of the axis of symmetry in the factored equation. Solve to find y.

Expanding Factored to Standard

You use the method of expanding when you want to go from factored to standard form. You multiply each term in the brackets, and then collect the like terms.

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Common Factoring

You can use common factoring when there is a number or variable which you can divide out evenly from both terms. When factoring, you are looking for the GCF of the coefficients and the GCF of the variables.
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Simple Factoring: x² + bx + c = (x + r)(x + s)

When given an equation in standard form, you can factor to get factored form, by using the product and sum method. This is where you find 2 numbers whose product is (c) in the Standard form, and if those same numbers are added, they should equal to (b). Here's a quick video explaining this method.
Example 1: Factoring quadratics with a leading coefficient of 1 | Algebra II | Khan Academy

Special Cases

A polynomial of the form a² - b² is a difference of squares and can be factored as

(a - b)(a + b), and a polynomial of the form a² x² +- 2 a b x + b is a perfect square and can be factored as (a x +- b)².

Quadratic Special Case's

Factor by Grouping

In this method, you factor out common numbers or variables, and collect the like terms
Factor expressions by grouping

Vertex Form: a(x - h) + k

When an equation is in vertex form, a(x-h)² +k each letter is responsible for a transformation:



  • The (-h) moves the vertex of the parabola left or right , when the (h) is negative it moves to the right and when its positive it moves to the left.
  • The (k) moves the vertex of the parabola up or down, when the (k) is negative it moves down and moves up when its positive.
  • The (a) vertical stretch or compression. The parabola is reflected upon the axis if a the (a) is negative.



When solving for the x-intercepts set y=0 and solve the equation. There could be 2 solutions, 1 solution, and no solution.


The vertex is given by the equation, where (h, k) is the vertex.

Standard Form: ax² + bx + c

In this equation:



  • (a) gives the shape and direction of opening of the parabola
  • (c) is the y-intercept


You can solve for the x-intercepts using factoring, completing the square, or the quadratic formula.


The Quadratic Formula solves for the x-intercepts

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Discriminant: is a part of the quadratic formula which determines how many solutions there are to an equation.
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To graph the parabola in standard form you need to:
Find the X-Intercepts:
  • Use the quadratic formula to get the zeros.
Find the axis of symmetry:
  • Add the two solutions together and then divide by 2 to get the axis of symmetry.
Find the vertex:
  • Substitute the (x) value you solved for above, and solve for y.
  • The axis symmetry will be the (x) value in the vertex
  • Vertex: (h,k) h=x value, k=y value
Completing the Square: from standard form to vertex form, you need to complete the square. Completing the square is very simple. You need to take the (b) value and divide it by 2, once you get that answer you square it. This video shows you the exact steps you need to make a perfect square trinomial.
Completing the Square - Solving Quadratic Equations

Word Problems

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Link Between Equations and Graphs

Equations and Parabolas: All three forms of equations, describe a parabola. It gives you information about different parts of the parabola, you either have to solve for it, or just by looking at it you can get some details. Equations are a written version of a graphed parabola, the graphed parabola is a visual for the equation. In each equation you have to solve for x-intercepts, and the vertex, this allows you to plot 3 points of the parabola.


Discriminant and X-intercepts: Discriminant's tells you how many solutions you have to an equation or parabola. This helps us find the exact number of x-intercepts.

Reflection

Quadratic Relations was a very interesting unit to learn. I found it easy the first few days, when we started this unit, but as we moved on it got confusing. When we began to learn about the different factoring methods, I was confused on what to use, when. I lost track of which method implies with what. As we went on with the course, I was able to ask questions, and understand what I was doing wrong. I have a better understand on how to find x-intercept, vertex of a parabola, and most importantly I know how to graph a parabola. By the end of the unit, I understand the basic concepts of quadratic functions.
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I did very well on the Quadratics Standard form test. I found this very clear and straightforward. Unlike the factored form, where the different methods had me confused. I felt very comfortable with the Quadratic Formula, which I thought was very helpful to find the zeros, and know how many there will be. Perfecting the square, of using the product and sum method was also something I found easy to understand, and use. Creating the equation and gathering information from the given equations, was also something I had liked. Overall this unit was very interesting and fun to learn about. I have learnt many new things, and added on to what I had learnt in grade 9. It was an amazing experience.