Quadratics

By : Azan B.

Table of contents

Introduction

What is it

The Algebra Of Quadratics

Quadratic Function

Analyzing Quadratics (Second Differences)

The Parabola

Transformations


Types of Equations

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

Vertex Form a(x-h)²+k

Standard Form ax²+bx+c


Vertex form

Finding an equation when only given the vertex

Graphing vertex form


Factored Form

Factors and zeroes

multiplying out brackets and quadratic expressions

Common factoring

Factoring simple trinomials

Factoring complex trinomials

Perfect square

Factoring by grouping

Graphing factored form


Standard form

Completing the square

Quadratic formula

Graphing from standard form


Word Problems

Revenue

Numeric


Reflection

INTRODUCTION

What is it?

The prefix quad- means “four” and quadratic expressions are ones that involve powers of x up to the second power . So why are quadratic equations associated with the number four that's because these equations are personally connected with problems about squares and quadrangles.Usually this stuff leads too quadratic equations. For example, a problem like: A quadrangle has one side four units longer than the other. Its area is 60 square units. What are the dimensions of the quadrangle? If we make the length of one side of the quadrangle as x , then the other must be x + 4 length. We must solve the equation x(x+4)+60 , it is equivalent to solving the quadratic equation x^2 +4x-60+0 Solving quadratic equations.

The algebra of quadratics

An expression of the form ax²+ bx+ c with x the variable and a, b, and c fixed values is called a quadratic. To solve a quadratic equation u have too solve an equation that can be written in the form ax²+ bx+ c=0 .

Quadratic Function

Substitute into and evaluate linear and quadratic functions represented using function notation, including functions arising from real-world applications. Through investigation using technology, the roles of a, h, and k in quadratic functions of the form f(x)=a(x – h) 2 + k, and describe these roles in terms of transformations on the through investigation with and without technology,from primary sources, using a variety of tools, or from secondary sources, and graph the data.

Analyzing quadratics (Second Differences)

While the first derivative can tell us if the function is increasing or decreasing, the second derivative tells us if the first derivative is increasing or decreasing. If the second derivative is positive then the first derivative is increasing, So that the slope of the line to the function is increasing as x increases. We see this phenomenon graphically as the curve of the graph being concave up, that is, shaped like a parabola open upward. Likewise, if the second derivative is negative, then the first derivative is decreasing, so that the slope of the tangent line to the function is decreasing as x increases. Graphically, we see this as the curve of the graph being concave down, that is, shaped like a parabola open downward.
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Parabola

Parts of a parabola

vertex- is the highest or lowest point of the parabola

Line of symmetry- The very middle of the parabola through the x axes

x-intercept - Is where the parabola crosses the x axes

y-intercept - is where the parabola crosses the y axes

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Graphical Transformation Of Functions

For Graphical Transformation of Functions you use f(x) = a(x − h) 2 + k and you graph it. The graph of the quadratic function is shaped like a letter U and is called a parabola. Each of the constants in the vertex form of the quadratic function plays a role. The constant h controls a horizontal shift and placement of the axis of symmetry, and the constant k controls the vertical shift. Let’s begin by looking at the scaling of the quadratic.

Scaling The Quadratic

The graph of the basic quadratic function f(x) = x 2 is called a parabola. The following table shows what plots are to be plotted to get the parabola. Also to get the other side of the parabola you must plot the point on the opposite side.
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Types of equations

Vertex form: a(x-h)²+k


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


Standard form: ax²+bx+c

Factored form

y= a(x-r)(x-s) is the equation of a factored form.


You have to multiply everything within the brackets by each other before multiplying it by "a".


(x)(x) (x)(s) (r)(x) (r)(s)

After multiplying these you then simplify and then multiply it all by the a value


This is called expanding and simplifying

3.1 Analyzing Quadratics

Vertex form

The vertex form of a quadratic function is when you take f (x) = a(x - h)2 + k and (h, k) is the vertex of the parabola.


For example if an equation is 0=4(x-2)3+5 the vertex will be (2+3)


How did it get from -2 to 2 well that's because the x coordinate is always changed to a positive if the number is a negative or switched to a negative if the number is already a positive.

Standard form

It can be used to write a quadratic expression in an alternative form.

Vertex form

Finding an equation when only vertex form is given

3.4 Finding the Equation given Vertex

Graphing from vertex form

Start with the function in vertex form:


y = a(x - h)2 + k ( y = 3(x - 2)2 - 4)


Pull out the values for h and k.


If necessary, rewrite the function so you can clearly see the h and k values.

(h, k) is the vertex of the parabola.


Plot the vertex. y = 3(x - 2)2 + (-4)


h = 2; k = -4


Vertex: (2, -4)

3.2 Graphing from Vertex Form

Factored form

Factors and zeros

Rewrite the factors to show the repeating factor.

( x - 3)( x - 3)(2x + 5)

Write factors in polynomial notation. The a is the leading coefficient of the polynomial

P(x) = a( x - 3)( x - 3)(2x + 5)


Multiply ( x - 3)( x - 3)


Then multiply ( x 2 - 6x + 9)(2x + 5)


For right now let an = 1


The polynomial of lowest degree with real coefficients, and factors ( x - 3)2 (2x + 5) is: P(x) = 2x 3 - 7x 2 + 8x + 45.



The main concept is simply to multiply the factors in order to determine the polynomial of lowest degree with real coefficients.

Multiplying out of brackets and Quadratic expressions

Starting with the quadratic expression x^2+5x+6, we can carry out a process which will result in the form (x + 2)(x + 3) This process is called factorizing the quadratic expression. For example, if we wanted to solve a quadratic equation. So the equation is formed when we set a quadratic expression equal to zero, as in x^2 + 5x +6=0 This is because the equation can then be written (x + 2)(x + 3) = 0 and if we have two expressions multiplied together resulting in zero, then one or both of these must be zero. So, either x + 2 = 0, or x + 3 = 0, from which we can conclude that x = −2, or x = −3. The solutions of the quadratic equation is x^2 + 5x + 6 = 0.

Common factoring

Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved


Finding the GCF of a List of Integers or Terms

1)Prime factor the numbers.

2)Identify common prime factors.

3)Take the product of all common prime factors.

If there are no common prime factors, GCF is 1

Factoring simple trinomials

What i like to do for this type of factoring is just use mental math but others may find it difficult. To factor a simple trinomial you need to find 2 numbers that multiply to give you the C value but also add up to give you the B value


For example:


x^2 + 7x + 6


6 and 1 are factors of 6 that will equal 6 if multiplied and 7 if summed.


You will then use these 2 numbers part of the equation y=a(x-r)(x-s)


y=(x+6) (x+1)

Simple Trinomial Factoring

Factoring complex trinomials

This type of factoring is a bit confusing so I'll break it into steps.


1. You first need to multiply the A value with the C value.

2. Next, find the common factor of the product you get and find 2 numbers that will add together to give you the B value

3. After you find the 2 numbers sub them into the equation and group them.

4. Find common factors of each group and you will then use the numbers inside the brackets as well as the 2 numbers you got from grouping.

Perfect Square

A perfect square is a trinomial, in which the first term and third term have a square root. To get the second term we multiple the square roots of the first and third term then double it. That is how we get a perfect square.
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Factoring by grouping

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Graphing factored form

A quadratic equation is a polynomial equation of degree of 2. The standard form of a quadratic equation is 0 = ax^2 + bx + c


where a, b, and c are all real numbers and a 0.


If we replace 0 with y , then we get a quadratic Function


y = ax^2 + bx + c


whose graph will be a parabola.

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Standard form

Completing the square

COMPLETING A SQUARE

Quadtratic Formula

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Algebra Help - The Quadratic Formula - MathHelp.com

Word problems

Revenue

Calculators are sold to students for 20 dollars each. Three hundred students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?

Let R represent Revenue

Let X represent increase or decrease

To solve this word problem we have to use a certain equation (Price)(People)


That gives us

R=(20+5x)(300-30x)

R=-150x² + 900 x + 600


Next we have to complete the square which gives us a Maximum Revenue of $7350 and a selling price of $35

Numeric

The sum of the squares of two consecutive integers is 365. What are the integers?


Let one number be x. Since the two numbers are consecutive, the other number is x + 1.

The sum of the squares is (x)^2 + (x + 1)^ 2 and it is equal to 365. The equation is (x) 2 + (x + 1)^2 = 365.


If you look at the question, it does not say maximum, minimum so we do not need the vertex. What we need to do, is to solve the equation (x)^2 + (x + 1)^2 = 365, by factoring or quadratic formula.


Expand and simplify the equation into standard form: 2x^2 + 2x – 364 = 0 Factor or use the quadratic formula to find the two solutions: x = -14, x = 13.


The two solutions to the equation will give us two possible sets of integers which are answers to the question: The first set of integers is -14 and -13, and the second set of integers is 13 and 14.

Reflection

The unit of Quadratics was one of the hardest subject i have done so far in Grade 10 math and i hope to improve on it a lot more because once you understand the unit it becomes so much fun to do and seeing how you get answers in quadratics makes me feel smart. Quadratics is a really fun subject once it is properly understood. If i were asked to solve a problem quadratics related i would be very interested in wanting to do it. I don't have much tests and worksheets related to quadratics but we did do a total of 3 tests and i didn't do so well on 2 of them but the last one im pretty confident in my mark because i properly understood the unit and how to do it. Overall this unit was really hard and required a lot of hard work to do it and also was a lot of fun and in future math i would love to try this again