By: Manmeet Chatwal
Summary of website
Table Of Contents
TABLE OF CONTENTS
Understanding a Parabola
-What is a parabola?
-Analyzing a parabola (second differences)
Types of equations
- Understanding the step pattern
- Describing Transformation
- Graphing vertex form x
- Isolating for a variable
-Finding optimal value
-Finding axis of symmetry.
What Is A Parabola
Parabola use in real life
Properties of a parabola
FIND OUT IF AN EQUATION IS QUADRATIC
when using a table of values, you can then tell if the equation is quadratic or not. A similarity all quadratic equations have is that the y values have the same second differences , and the first differences are unequal; compared to linear equations which have the same first differences.
VERTEX FORM EXAMPLE EQUATION
FINDING AXIS OF SYMMETRY IN VERTEX FORM
To find the axis of symmetry (y=a(x-h)+k) in a vertex form equation you need to look at the h value as explained briefly before. If the (h) value is positive you will simply need to change the sign, and what the value then becomes is the axis of symmetry.
In the equation the value of h is +3, so when the sign is flipped the axis of symmetry is -3.
THE STEP PATTERN
The step pattern is used to create the u-like shape of the parabola; the step pattern is as follows
Over 1 up 1
Over 2 up 4
Over 3 up 9
Note: *When using the step pattern we just square the number we are going over by to find the number we go up by
Note: *When there is a whole positive/negative: number, fraction, decimal in place of variable a value you just simply multiply the a value to the squared part of the step pattern
BREAKING DOWN AND UNDERSTANDING THE EQUATION & TRANSFORMATIONS
After you plot your vertex using the transformations explained you can use the step pattern to plot your vertex form equation!
-The Y variable represents the Y-intercept
-The A variable represents whether the parabola opens up or down (positive a value results to parabola opening down, negative a value results to the parabola opening down), it also represents if the parabola will have a compression or stretch
- The X variable represents the x intercept
- The H variable represents where the X point is on the graph (right or left) (axis of symmetry) (If h is positive 6 the parabola will be horizontally translated 6 units to left)
- The ^2 creates the arch shape of the parabola
- The X variable is in charge of whether the parabola will be vertically shifted up or down (according to positive or negative sign) (positive K value will shift parabola up and negative K value will shift parabola down)
With all variables at 0 the parabola would be at the origin at 0,0
When a equation is in vertex form each letter variable is in charge of some type of transformation.
- The (-h) represents whether the parabola moves left or right. *RECORD (When the h value is positive the parabola moves left and when the h value is negative the parabola moves right). This is called a horizontal translation.
- The (a) value stretches or compresses the parabola. If the (a) value is greater than one the parabola will stretch by the factor of the worth of the variable; This is what is called a vertical stretch. If the (a) value is less than one the parabola will compress by the factor of the worth of the variable; this is called a vertical compression.
Also, the H value and K value make up the vertex
- The (k) variable represents whether the parabola moves up or down. *RECORD (When the k value is positive the parabola moves up, when the k value is negative the parabola moves down). This is called a vertical translation.
FINDING OPTIMAL VALUE OF A STANDARD FORM EQUATION.
Therefore the k value is positive 4, meaning the optimal value of the parabola will be +4 over the x-axis.
FINDING X INTERCEPTS/ZEROS in vertex form
put in an example for finding x intercepts here
GRAPHING VERTEX FORM
In factored form we will learn how to factor a variety of equations into factored form equations; throughout the unit this part is the most tough but most important topic of the unit. It is very important that you learn this part properly and you completely understand it!
common factoring: is when you find the factors or two or more numbers , and they both get some factors that are the same, which means they are common
expanding: expanding means we remove the brackets and what ever is inside them by multiplying, which is called the parentheses
grouping: means to group terms with common factors before factoring
simple trinomial: a trinomial is a polynomial with 3 terms
FINDING ZEROES IN FACTORED FORM
The zeroes of a parabola are called the parabola's x- intercepts, and to find the x- intercepts you want to use zero for the y-value, In other words you have to be compelled to take the co-efficient's within, and change the sign which would be your zero.
FINDING AXIS OF SYMETTRY IN FACTORED FORM
For example: X int @(8,0), (-6,0)
also the number one would be the Axis of symmetry because after you average the x intercepts 1 is the outcome meaning it is also the axis of symmetry.
FINDING THE OPTIMAL VALUE FOR EQUATIONS IN FACTORED FORM
STANDARD FORM= ax^2 + bx + c
THE QUADRATIC FORMULA
The quadratic formula allows us to solve for the zeros/x-intercepts and also allowing us to find the axis of symmetry by utilizing standard form equations. Following this further, if the equation was to be in a different form, it can simply just be expanded into standard form anytime, and then use the quadratic formula to solve for it. Below there is a picture of a quadratic equation, which we will also learn to apply.
SONG ABOUT THE QUADRATIC FORMULA
FINDING X INTERCEPTS OF STANDARD FORM EQUATIONS USING THE QUADRATIC FORMULA
FINDING THE AXIS OF SYMMETRY OF A STANDARD FORM EQUATION
example: let's find the axis of symmetry for the equation we found x-intercepts for above (4x^2-5x+1).
Method 1 (adding both x intercepts and divide by 2)
Axis of symmetry: (1+0.25)/2
Method 2- Using -b/2a (variables were identified the in previous method above)
Axis of symmetry: -b/2a
Therefore we used both methods to get the axis of symmetry of a standard form equation and we managed to get the same answer from both methods proving that our answer is correct.
FINDING THE OPTIMAL VALUE OF A STANDARD FORM EQUATION
With the example used above for explaining how to calculate x intercepts and axis of symmetry, we will also use the same example to explain how to find the optimal value.
Equation being subbed into: (4x^2-5x+1=0, X=0.625 (Axis of symmetry)
(I rounded 0.625 to 0.63 in example below just incase you never knew or got confused)
Therefore you have just figured out your optimal value using few simple and easy steps.
- negative, then there is no solution
- if it is equal to 0 there is only 1 solution
- if it is greater than 1 there is 2 solutions.
COMPLETING THE SQUARE
A CHART TO HELP YOU UNDERSTAND BETTER
FACTORING SIMPLE TRINOMIALS
FACTORING COMPLEX TRINOMIALS
DIFFERENT OCCURRENCES OF FACTORING (DIFFERENCE OF SQUARES AND PERFECT SQUARES)
EXPANDING FACTORED FORM EQUATIONS TO STANDARD FORM
revenue word problem video made by me
fencing word problem video mad by me
MY REFLECTION ON THIS UNIT
In conclusion you got to learn many different things during this quadratics unit. I will always encourage you to keep up to date by completing your homework and paying attention and asking questions in class if you do not understand one concept or another. I hope the concepts/terms I showed you through a variety of media use (pictures, text, videos) helped you understand this unit! I would like to thank you for your time!