Quadratics

By Ramanveer Brar

Table of Contents

Vocabulary

Linear or Quadratic

Parabolas

Graphing Parabolas Video

Factoring

Types of Forms

All About Vertex Form

The Axis of Symmetry and Optimal Value

Transformations
Zeros & X-intercepts

Step pattern

All About Factored Form

Zeros & X-intercepts

Axis of Symmetry

Optimal Value

All About Standard Form

Zeros

Axis of Symmetry

Optimal Value

Completing the Square to Turn to Vertex Form

Factoring to Turn to Factored Form

  • Common
  • Simple Trinomial
  • Complex Trinomial
  • Perfect Squares
  • Difference of Squares

Word Problems

Making Connections

Reflection

Vocabulary

Non-linear relation: a relationship between two variables that does not follow a straight line when graphed.

Curve of best fit: a smooth curve drawn to approximate the general path or trend in a scatter plot.

Parabola: the graph of a quadratic relation, which is U-shaped and symmetrical

Vertex: The point on a parabola where the curve changes direction. The maximum point if the parabola opens down. The minimum point if the parabola opens up.

Axis of symmetry: the line that divides a figure into two congruent parts.

Zero: a value of x for which a relation has a value of 0. Corresponds to an x-intercept of the graph of the relation.

x-intercept: the x-coordinate of the point where a line or curve crosses the x-axis. At this point y=0

Perfect square trinomial: a trinomial that is the result of squaring a binomial.

Difference of squares: an expression that involves the subtraction of two squares.

Linear or Quadratic

It is very easy to find out whether and equation is linear or quadratic. You must find the differences. If the first differences are the same then this equation is linear. If the second differences are the same, the relation is quadratic. Below is an example of a relation that is quadratic.
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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 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 y co-ordinate of the vertex
  • The y-intercept of a parabola is where the graph crosses the y-axis

To graph parabolas you can use Desmos. It is a quick substitute to drawing a parabola onto graph paper.

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Below is a video I made, i think it helps you understand how to graph parabolas in a simple and fast way.
Graphing Parabolas

Factoring

In order to really understand quadratics, how to make parabolas and how to convert forms you must know how to common factor properly. Factoring is a big part of quadratics. The video below will help you understand common factoring better.
Common Factors

Types of Forms

Vertex form - a(x-h)²+k

Factored form - a(x-r)(x-s)
Standard form - ax²+bx+c

Vertex Form

The vertex form is: y = a (x - h)^2 + k

Axis of Symmetry (x=h)

Optimal Value (y=k)


Transformations

Orientation

If the a value is greater than zero the parabola opens up.

If the a value is less than zero the parabola opens down.

Shape

If the a value is zero the parabola is vertically compressed.

If the a value is greater or less than one the parabola is vertically stretched.


Zeros & x-intercepts

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Step Pattern

The step pattern is the rate at which the parabola goes up/down.

Ex. up 1, up 3, up 5, up 7, each time it goes up more but to the side by 1. This is just one example.

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Factored Form

Zeros & X-intercepts

y = ( x - 3 ) ( x - 4 )

x - 3 = 0 x - 4 = 0

x = 3 x = 4


Axis of Symmetry

x = 3 + 4 / 2

x = 3.5


Optimal Value

y = ( x - 3 ) ( x - 4 )

y = ( 3.5 - 3 ) ( 3.5 - 4 )

y = 12.25 - 14 - 10.5 -12

y = -24.25

3.5 Graphing from Factored Form

Standard Form

Below you can see how to find zeros, axis of symmetry and the optimal value.
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Below is the step by step process of completing a square.
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3.14 Summary of three forms

Word Problems

Example 1
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Here is the solution...
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Example 2
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Example 3
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Making Connections

Everything we learned in the three units of quadratics tied together at the end. The three types of forms can be converted into each other. You really need to understand everything and how to put it together to be able to put things together. All forms can be graphed but you must know how to convert, and you should know the formulas. As you get further into this unit it gets more advanced but you may find shortcuts and easier ways to do a few equations.

Reflection

This was the second mini test for quadratics. I did not do very well, i could have done much better. This test was definitely more difficult than the first mini test because the first one was mostly based on parabolas, and how to find everything in it. Whereas this test was based on mostly equations, expanding, simplifying, factoring and there were more word problems and the communication question was hard as well. The communication section is where i did the worst, when i got my test back i realized that it was straight from the homework. I know that i have improved on factoring and writing it in the correct form.
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