Quadratic Relations

By: Harkomal Mann

Introduction

Quadratics are equations which once graphed create curves. They could be hard to understand sometimes but this website helps you get a better understanding. A goal for this unit is being able to understand the different ways to graph the vertex form.

Vertex Form

What is a Vertex?

Vertex is the highest point or lowest point on a graph​h. The vertex is where the graph changes direction. It is also the location where the axis of symmetry and parabola meet (h,k).

Axis of Symmetry

It is a vertical line that divides the parabola into two congruent half's. It always passes through the vertex of the parabola. (x=h).

Optimal Value

The optimal value is the minimum or the maximum value (the k value). It is the value of the y co-ordinate of the vertex. If the parabola opens upwards it will be the minimum value and if it opens downward then it will be the maximum value. (y=k).

Quadratic Relations

To calculate first differences, first you need to subtract the second y value from the first y value. You do this for every pair of y-values. When the first differences are constant it means that the pattern is linear. All x values must be in order. When the second differences are constant, the pattern is quadratic.

Identifying Transformation in Vertex Form

For a simple parabola with a vertex that is at (0,0) is y=x² which is used to make a new graph that has a corresponding new equation. This equation could be written in vertex form. Which is

y=a(x−h)2+k

  • a represents the direction of opening and compression or stretch (it is a stretch if the number is a whole number, if number is a decimal or fraction then it would be a compression).
  • k is the vertical translation (moves parabola up or down depending on it's sign. If negative it moves down, and if positive, it moves up).
  • h is the horizontal translation (moves parabola left or right depending on it's sign)
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The Zeros and Y-intercepts

Zeros are the x-intercepts or roots and it is where the graph crosses the x-axis (x,0). Y-intercept is where the graph crosses the y-axis (0,y).

Graphing Vertex Form

To graph in vertex form, first you need to identify the translations of a, h, and k. This is found in the equation. The two ways to graph vertex form is mapping notation and the step pattern.
Using the Mapping Rule to Graph a Transformed Function
This is how you graph using the mapping notation
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Word Problem

The height, h, in meters, of a soccer ball, t seconds after being thrown can be modeled by the equation: h= -0.10 (t - 4)²10

a) What is the soccer ball's height 3 seconds after it is kicked?

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The height of the soccer ball is 10 m.
b) What is the initial height of the soccer ball?
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c) what time did the ball reach its maximum height?

The ball reached it's maximum height at 3 seconds because of the x value of the vertex.

Additional Information

  • to find the y-intercept, set x=0 and solve for y
  • to solve, set y=0 and solve for x or expand and simplify to get the standard form, then use the quadratic formula

Factored Form

MULTIPLYING BINOMIALS & SPECIAL PRODUCTS

We use the FOIL Method when expanding two binomials


FOIL (First, Outside, Inside, Last)


An example would be:

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When there is a number in front of the two binomials, multiply the binomials first.
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Special Cases

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These equations above are perfect sqaures

Difference of Squares

To calculate to the difference between perfect square's is : (a+b)(a-b)= a2-b2
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Common Factoring

The GCF is the greatest number in common between two numbers.

4x+2


First: Find the GCF which is 2

Second: Write the solution with brackets 4x/2 + 2/2


Equations with exponents still have the same steps

An example would be:

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Reminder: Always check by expanding and simplifying.
The factor of an integer is any number that divides evenly into the integer.

For example 24 has factors : 1 2 3 4 24 12 8 6

When factoring polynomial expressions, we need to examine both the numerical coefficients and variables to find the greatest common factor.


Factoring is the opposite of expanding

Quadratic Equations in Factored Form

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

The zeros, the x-intercepts, the roots, the solution (to "solve")


  • Value of a gives the shape and direction of opening
  • The value of r and s give the two x-intercepts (r,0) and (s, 0)
  • To find the vertex, use the zeros to find the AOS and sub this x value into the given equation and solve for y

Simple Factoring

Algebra tiles example of factoring

A quadratic in standard form can factor to get you factored form


x^2+bx+c (x-r)(x-r)


Standard Form to Factored Form

Where r+s=b sum

and rs=c product

and r and s are the integers


Step 1: Look at signs of b and c in the given expression (x^2+bx+c)

  • If b and c are positive, then both r and s are positive
  • If b is negative and c is positive, then both r and s are negative
  • If c is negative and one of r and s is negative

Step 2: Find the "product and sum"

  • Find the two numbers whose product is c
  • Find two numbers which numbers sum b


Ex: x^2+4x-5

(x+5)(x-1)

If c is negative, one of r or s is negative

Complex Factoring

Complex trinomials have a coefficient that is other than 1 in front of the x^2 term.


Where a can never be 1.

Difference of Squares and Perfect Squares

How to factor 36x^-4

Recall: (a+b)(a-b)= a^2-b^2

Apply this concept to the question

To find a and b, simplify square root and first and last term.

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Common Factoring
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Perfect Square Trinomials

A perfect square trinomial is a trinomail which results from squaring a binomial. These trinomials can be factored by using the patterns from expanding bionomials
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Here are some examples:
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REMINDER: In perfect square trinomials, the following conditions meet, if not then the trinomials are not a perfect square.

(1) the first and last terms are perfect squares
(2) the middle term is twice the product of the square roots of the first and last terms

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Finding Zeros

To find the x-intercepts of a quadratic relation we have to put the equation into factored form and solve (set y=0)
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When you are solving a equation that needs to be factored, one side must always equal to 0.

An example of a simple trinomial...


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To check your answer,by first substituting both roots into the left side and right side of the equation of the original. Check one only.

Difference of Squares Trinomial

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Complex Trinomial

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Key Information

  • the value of a gives you the shape and direction of opening
  • the value of r and s give you the x-intercepts
  • to find the y-intercept, set x=0 and solve for y

Graphing

Graphing Parabolas in Factored Form y=a(x-r)(x-s)

Word Problem

The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h=-5r^2+20t+60 where t is the time in seconds, and h is the height in meters.


Write the formula in factored form

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Standard Form

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Maximum and Minimum Value

In order to find the maximum and minimum value, you need to complete the square.
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Here is an example...
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If the "a" value is a number and/or a negative number?

Step 1: Group the x terms together

Step 2: Common factor only the constant terms.

Step 3: Complete the square

Step 4: Write the trinomial as a binomial squared

Quadratic Formula

While factoring may not always be successful but the quadratic formula can always find the solution. In order to use the quadratic formula, the equation has to be in standard form.

Down below is the quadratic formula:

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The quadratic formula is easier to use.
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Math Standard Form Example

Word Problem

You are selling hot dogs for $1 and sodas for $0.50. You make $200 by the end of the night. Let x represent the number of hot dogs sold and y represent the number of sodas sold. Write an expression to find out how many of each were sold.

Ax+By=C

1X+0.50Y=200

REMEMBER

The value of a gives you the shape and direction of opening and the value of c is the y-intercept.

Reflection

When we started quadratics I found it difficult. I did not quite get the concept but after continuing to practice it, it got easier. In the beginning I struggled on the tips but after a few weeks I started to understand the concept of quadratics more. During the word problems I feel like I struggled the most. It was hard for me to apply what I learned into word problems but I understood after I got some help. My favorite part about quadratics was the quadratic formula. I found this the easiest to understand. All in all, I think quadratics is a easy unit but you need to practice because it is challenging at the same time.
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