Quadratic Relations

By: Simran Bedi

What is quadratics?

  • Quadratic equations are of order 2
  • They only have 1 minimum or maximum value at the vertex
  • Maximum of 2 roots/zeroes
  • 1 y-intercept
  • 3 different types of equations:
  1. Vertex Form --> y= a(x-h)^2 +k
  2. Factored Form --> y=a(x-r)(x-s)
  3. Standard Form --> y=ax^2+bx+c
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Where is quadratics used?

Quadratic equations may seem like a non beneficial concept outside of the classroom, but they're actually used in everyday life. They are used on a daily basis to calculate anything from prices to areas to speeds. The equation can be used to find shapes, circles, ellipses, parabolas, etc. For example, any type of object with a curve (volleyball, tennis ball, football).


The quadratic equation is used by car makers to determine the amount and type of brakes needed to control a car going at numerous speeds. This is included in the design steps for new cards, motorcycles, trucks and many other automobiles.


The quadratic equation is used in the design of almost every product in stores today. It is used to determine the life expectancy of safety of the products being sold. This is useful to designers because it helps them see what is needed to be changed.


The quadratic equation has applied in various types of employment in the world beyond school.

Table of Contents

1. Vertex Form: y=a(x-h)^2+k

  • Axis of symmetry (x=h)
  • Optimal Value (y=k)
  • Transformations
  • X-intercepts or zeros
  • Step Pattern


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


  • Zeroes or x-intercepts (r and s)
  • Axis of Symmetry (x=(r+s)/2)
  • Optimal Value (sub in)


3. Standard Form: y=ax^2 +bx+c


  • Zeroes (quadratic formula)
  • Axis of Symmetry (-b/2a)
  • Optical Value (sub in)
  • Completing the square to turn to vertex form
  • Factoring to turn to factored form
> Common
> Simply Trinomial
> Complex Trinomial
> Perfect Squares
> Difference of Squares

VERTEX FORM


  • To find x-intercepts (zeros):
  • Let y=0 and solve for x
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Important Points

  • Lowest or highest point (depending on the direction)
  • The quantity "x-h" is squared, therefore, it's value is always zero or greater, being squared, it can never be a negative

Step Pattern

1. Find vertex


2. Count units to the right to get to the next point (horizontal translation)


3. Count units going up to the next point (vertical translation)


4. You should be at the location of the next point.


Vertex > Next Point > Next Point

> 1 Right > 1 Right

> 1 Up > 3 up

Graphing Quadratics Using Step Patterns

Strategies

When graphing a vertex form parabola:


  • Check the sign a to see if it is opening up or down
  • Find the vertex and y-intercept
  • Determine if there are any x-intercepts
  • Check if there are enough points to finish the graph

FACTORED FORM

Zeroes

  • A zero of a parabola could also be known as x-intercepts
  • To find the x-intercept, y must = 0
  • i.e: (x,0)


  • When the (a) value changes, the axis of symmetry do not change
  • When the (a) value changes, the zeroes do not change
  • When the (a) value changes, the optimal value does change

  • The given equation in factored form can easily find the roots


e.g:

y=(x+3)(x-1)

0=(x+3) (x-1)

x+3=0

x-1=0

x=-3

x=1

Axis of Symmetry

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Optimal Value

  • Is the y-value of the axis of symmetry
  • Lowest or highest value
  • To find the optimal value, find the axis of symmetry and plug it in to the equation for x -- solve for y

e.g - A of S: x=1

y= (x-3)(x+5)

y= (1-3)(1+5)

= (-2)(6)

= -12

- Together, the axis of symmetry and optimal value give the vertex (1,-12).

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STANDARD FORM

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3 15 Quadratic Formula

Discriminant

The discriminant is the component within the quadratic formula which can determine the number of solutions an equation has.
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Axis of Symmetry

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Optimal Value

  • Plug in the value of your axis of symmetry for x and then solve for y
  • If you can find your roots and vertex, then you can graph
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Completing the Square

  • The process where you change a standard form equation in to a vertex formed equation


  1. Factor the coefficient a if possible and place it outside of the parenthesis
  2. Divide coefficient b by 2 and then square it
  3. Add and subtract the value you get (you can't just add a number, so you have to subtract it as well)
  4. Simplify the equation by combining the terms at the end

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Completing the square

FACTORING

Common Factoring

  • To factor a number or expression, it simply means to write it as multiplication
  • Product of factors


Example: Factor 30

30= 5 x 6 = 5 x 2 x 3


Factor 50

50 = 2 x 25 = 2 x 5 x 5


Factor 3x-3y

= 3(x-y)

Simple Trinomial

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Complex Trinomial

  1. Multiply the lead coefficient by the lead term
  2. Find the two numbers that make the product from step 1, but add to make the middle term coefficient
  3. Re-write the original trinomial, replacing the middle term with two terms whose coefficients are the numbers from step 2
  4. Common factor the first two terms from step 3
  5. Now common factor the last two
  6. The common factors do not match, but the brackets do; put the common factors in their own bracket
  7. Lastly, FOIL out your answer
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Perfect Squares

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Difference of Squares

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Word Problems

3.12 Motion problems

Number Problem

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Geometry Problem

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Economic Problem

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Connections Between Topics

Graphing and Equations:


Vertex, standard and factored are all three forms that can be graphed. To be able to plot three points, you will need to find your x-intercept (zeroes). In the equation, you can find the y and x value from the vertex. By looking at your equation, you can get an image of what your parabola will look like on the graph (concave up or down).

REFLECTION OF THE UNIT

When we first started this unit, I found it to be pretty straight forward. I was able to understand the concept of a parabola and successfully graph them too. As we kept going further on in to factoring, that is when I started to get confused. I found it difficult because we would learn something new everyday; keeping track of all the different methods was not easy. I feel like I struggled with not knowing which method to use because of all the different options we had. As we started to get quizzes in class, I never got the greatest marks. I know that if I practiced more and came in for more help, I wouldn't have struggled as much. A test that I was pleased with was the Quadratics - Test 1. I did well on everything but communication. After receiving this test back, I knew that I had to work on my communication skills, and read the questions given to me more properly.


In my opinion, completing the square is one of the hardest things I've learned. I still need practice with it right now because I get confused with the steps and what to do next. After moving on and being taught about the quadratic formula, I would have to say that I actually enjoyed learning about it. I find that it is simply and I am confident in answering any question related with the formula. In my opinion, the world problems are confusing and sometimes complicated to do. Every time I am confused, I always watch Mr. Anusic's math videos, which I am very thankful for. They're extremely helpful, and when I'm confused at home, I just watch his videos, which is very convenient. Overall, I think I'm doing okay in quadratics, but with a little more practice, I feel like I would be more satisfied.

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