Quadratics

By: Nargiz M

Table of contents

  1. Introducing the Parabola.
  2. Analyzing quadratics
  3. Transformation
  4. Graphing from vertex form
  5. Finding the equation given vertex
  6. Graphing from factored forming
  7. Expanding
  8. Common factoring
  9. Factoring by grouping
  10. Factoring simple trinomials
  11. Factoring complex trinomials
  12. Factoring special trinomials
  13. Completing the Square
  14. Solving by isolating x
  15. Discriminant
  16. Quadratic Formula
  17. Shape problems
  18. Revenue problems
  19. Motion problems
  20. Consecutive intergers
  21. Reflection

What is Quadratics?

Lesson #1: What is a Quadratic Function?

The graph of a Quadratic Function is a curve better known as a Parabola. Parabolas may open upward or downward and vary in width and steepness. However, they all have the same basic curved “U” shape. The picture attached below show four detailed graphs, all of which are Parabolas.

All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point known as a vertex of the parabola. If the parabola opens up, then the vertex is the lowest point, and is known as the minimum point. In both, the vertex is the optimum value. – the minimum and maximum points on a graph- If the parabola opens down, the vertex is the highest point. This point is called the maximum point. A parabola also contains two points called the zeros better known as the x-intercepts

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Table of Values to analyze Quadratics

Lesson #2: Using a Table of values to analyze Quadratics

As previously learned, Linear systems are straight lines, Linear relations the first differences are always the same, however in Quadratic Functions the second differences are the same.

A quadratic function can be graphed using a table of values. The graph creates a parabola. The parabola contains specific points, the vertex, and up to two zeros or x-intercepts. The zeros are the points where the parabola crosses the x-axis.

If the coefficient of the squared term is positive, the parabola opens up. The vertex of this parabola is called the minimum point. Example : y = a(x-h) + k


If the coefficient of the squared term is negative, the parabola opens down. The vertex of this parabola is called the maximum point. Example : y= -a (x + h) +k

To help you graph, Technology can be very helpful, try using desmos , to help you graph parabolas.

To graph a parabola you also need to know how to use a step pattern :
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Vertex Form

Vertex Form

The equation of a parabola can be expressed in either; Standard form and Vertex form.

The vertex form of a parabola is expressed as; y = a(x-h)2+k

A, H, and K are all essential factors of the Parabola.

A:

- If A is positive then the parabola opens upward

- If A is negative the graph opens downward

- If the value of A is greater than 1 then the parabola widens

- If the value of A is less than 1 then the parabola narrows down.

H:

- When translating the graph, H determines whether the Parabola will move sideways

- The coefficient of H is always opposite of what it is. + = neg , neg = +

K:

- When translating the parabolic graph, K determines whether the parabola will move up or down.


Here are the steps to graph a parabola only given the vertex form:

Step 1:

Find the vertex. Since the equation is in vertex form, the vertex will be at the point (h, k).


Step 2:

Find the y-intercept. To find the y-intercept let x = 0 and solve for y.


Step 3:

Find the zeros. To find the x-intercept let y = 0 and solve for x.


Step 4:

Graph the parabola using the points found in steps 1 – 3.


Example:

Y= a(x-h)^2 +K ---- (The vertex is (-2,8) and passes through the point (2,0)

Step One: insert all numbers into designated places. You will be solving for A.

y=a(x+2)^2 + k

y=a (x+2)^2 + 8

0 = a (2 +2)^2 + 8

0= a (4)^2 +8

0= a16 + 8

subtract 8 from both sides

-8 = a16

-8/16a = -1/2

a= -1/2

= y= -1/2 (x+2)^2 + 8


Now choose points, using the table of values, and graph the Parabola

Factored Form

Lesson: #6 Factored Form

The factored form of a Quadratic question is : y= 0.5(x+3) (x-9)


Step 1 State the zeroes : The Zeros are found by setting each factor equal to zero.

: Set y=0

x+3 = 0 x-9=0

(x=-3) x=9


Step 2 Axis Of Symmetry: is the midpoint of the zeroes.

X= -3+9 = 6/2 = 3

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Step 3 Optimal Value : is found by subbing the axis of symmetry value into equation

Y= 0.5 (x+3)(x-9)

0= 0.5 (3+3)(3-9)

=0.5(6)(-6) = -18 (Vertex)


https://www.youtube.com/watch?v=ZS3wZBBz5xM

The video attached will guide you into solving what is needed for this form.

Expanding

Expanding is very important in the quadratics unit; here is a note :
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Common Factoring

Common factor is a number that will divide into two numbers evenly.
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Factoring By Grouping

Factoring by grouping means that you will group terms with common factors before factoring.
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Factoring Simple Trinomials

Always remember to find the multiples of x2 and C, Add them and get the middle number.
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Factoring Complex Trinomials

Recall simple trinomials which start with a number that is not greater than 1.

But now… Complex trinomials starts with a number that is not equal to 1.


Remember to always find multiples of ax2 and C to get the middle term, also to common factor if necessary.

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Factoring Special Trinomials

Perfect squares -

Step 1: Square root and find the multiples of a and c

Step 2: Then multiple the numbers by 2 to equal middle term

Step 3: equals to middle term = perfect square


Different of squares - ( x + z ) ( x - z )

Step 1: Square root and find the multiples

Step 2: Insert both the numbers you got in the brackets one + and the other -

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Completing the Square

Moving from Standard y= ax2 + bx + c form to Vertex form Y= a(x-h)2 + k


Example :

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Solving by isolating x

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Discriminant

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Quadratic Formula

This formula helps you find the X-Intercepts

Step 1. Find the Quadratic Formula

Step 2. Substitute in the variables, So, plug in the values for a, b, and c

Step 3. Solve

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Quadratic word problems - Shapes

Question : A photograph measuring 12cm by 8cm is to be surrounded by a mat of uniform width before framing. The area of the mat is to equal the area of the photograph. Find the width of the mat.


Answer below :

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Quadratic word problems- Revenue

Question : Calculators are sold to students for 20$ 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?


Answer below :

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Here are some more revenue problems :
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Motion Problems

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Consecutive Integers

Consecutive means in sequence. In the context of numbers it means that the numbers differ by 1. For example, 8 and 9 are consecutive numbers, but 8 and 10 aren't.


Example of a Consecutive integer problem :

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Here is a video on Factoring to develop your understanding :

- Video includes the following

- Common Factoring

- Factoring By grouping

- Simple trinomials

- Complex trinomials

- Perfect/Different squares

Factoring

Reflection

In the Quadratics unit, we learned a LOT! Some of the main topics we covered were quadratic functions, standard form, factoring, graphing, completing the square, quadratic formula, word problems etc. I struggled a lot with the tests and assessments given during the quadratics unit. What i could've done to improve myself in this unit was to practice regularirly , study on my own a few hours a day and make it a habit. Also to do as much problems as i can especially study hard on the areas i have difficulties understanding and ask for help.
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This was quadratics 1 mini test, i found it easy to graph parabolas!
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This was one of the questions i did on my quiz, i did not common factor because i forgot. Next time i should pay attention to the questions/equations carefully and make sure i am not forgetting any steps.

Links Between Topics In Quadratics

Factoring connects to graphing

Factoring connects to graphing because when you solve and find the x-intecerpts of an equation you would know what each variable does and goes when graphing.

Simple Trinomials connects to Complex trinomials

Simple and Complex trinomials have the same equation the same way, except in simple trinomials the a-value equals to 1, and complex trinomials equal more than 1 for a-value.

Standard form connects to Vertex form

When completing a square both topics link together, because you convert standard form to vertex form in order to get the vertex. From ax2+bx+c to y=a(x-h)2+k

PEACE.