Quadratic Relationships

By : Bhargav

What is quadratics you ask?

Quadratics is an example of a Quadratic Equation. Quadratic Equations in particular make mathematical curves, that can not only be measured but also graphed. Quadratics is used for a number of things including making a design for bridges, flight paths for planes, it is also used for businesses (Finding trends) and it is also used in rollercoasters.

Here we have a few examples of quadratics being used.

What is a Parabola and a Hyperbola?

Parabola: A parabola is a quadratic function. It is graphed by a symmetrical curve and graphed on a 2 dimensional plane or X and Y sheet or plane. A parabola has either 1 or 2 x-intercepts. An equation or graph or nonlinear line that has only 1 x-intercept it's y value will always be 0 and the x value can be anything. There many different parts to a parabola such as the vertex, the axis of symmetry, Min/Max value, the y-intercept & the x-intercept(s) (Some parabola's have 1 x-int.)


Hyperbola: A hyperbola is quite similar to a parabola, but it as no x-int. This is because the line is "hovering in midair". It is graphed by a symmetrical curve and graphed on a 2 dimensional plane or X and Y sheet or plane. Despite having no x-int it will always have a y-int. There many different parts to a hyperbola such as the vertex, the axis of symmetry, Min/Max value and the y-intercept.

Factored Form

Learning Goals

Learning Goals

By the end I will be able to:
1) Identify what each variable represents.
2) Solve using Factors.
3) Use each and every type of factoring there is.

Summary of the unit

In this unit you will learn how to Complete the square, use the Quadratic Formula, do revenue questions and you will learn how to find the Max or Minimum value and the vertex. Then you will learn what each variable does and learn how to do a word problem.

Meaning of each variable in: 𝑦=𝑎(𝑥−𝑟)(𝑥−𝑠)

1st E.x. y=2(x-3)(x-1). The a value tells us that this equation is stretching by a factor of 2 and opening upwards. The x variables can be time or distance in an equation. The r and s variables give us the x-intercepts for the parabola, when using 3 point method.
2nd E.x. y= -2(x-1)(x+9). The a value in this equation tells us that it is stretching by a factor of 2 and is opening downwards. The x variables can be time or distance in an equation. The r and s variables give us the x-intercepts for the parabola, when using 3 point method.
3rd E.x. y= 0.5(x+2)(x+7). In this equation, a tells us that it is vertically compressing by a factor of 0.5. The x variables can be time or distance in an equation. The r and s variables give us the x-intercepts for the parabola, when using 3 point method.

Types of factoring:

Greatest Common Factor

Greatest Common Factor

How to do Common Factoring
You have this equation: 21x^2 + 42x
First find the greatest common factor, which is 7.
Then find the greatest common variable(s). which in this would be x
Next write out the Greatest Common Factor and the Common Variable.
In this case the answer would be 7x(3x+6)

Simple factoring (a=1)

Simple factoring (a=1)

How to do Simple Factoring
E.x. there is an equation: x^2 + 4x + 3
First find 2 numbers that multiply to a*c and adds to b. In this equation it would be 1 and 3
Then substitute these numbers in for 4x. It would be in this equation x^2 + 1x + 3x + 3.
Next common factor the first 2 terms and the last 2 terms. it should be like so x(x+1) + 3(x+1).
Take the numbers in the brackets and put only one. After that write the numbers and variables that are outside, in brackets.
The answer should be: (x+3)(x+1)

Complex factoring

How to do Complex factoring
E.x. you are given this equation: 3x^2 + 5x + 2
First multiply the first and last terms to find a number, then find 2 numbers that multiply to a*c and adds to b.
When found write them down in place of 5x, then common factor the 1st and 2nd terms along with the 3rd and 4th terms.
Then if there is a common in both brackets write one down and use your common factors to write the other part.
The answer to the above question is: (3x+2)(x+1)

Difference of squares

Difference of squares

How to do difference of squares
E.x. you have this equation: x^2 16
First square root this: x^2, which will give you just x
Next square root 16, which will give you 4
Finally write the answer out as (x+b)(x-b).
For this question it would be written as (x+4)(x-4)

Perfect square (a+b)^2

How to do a perfect Square.
E.x you have this: (a+5)^2
First write it out like so (a+5)(a+5)
Second rewrite the equation like this: a^2 + 2ab + b squared
Lastly solve.
video notes of all types of factoring

Word Problem

The path of a soccer ball is modeled by the equation: y= -1/16(x-28)^2 + 49. Where d is the horizontal distance, in meters, after it was kicked, and h is the height in meters above ground.

a) What is the maximum height of the soccer ball?

b) What is the horizontal distance when this occurs?

c) What is the height of the ball at a horizontal distance of 15m?
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