Learning About Quadratics

Fun with Quadratics!

What is a Quadratic?

A quadratic is a variable that gets squared

QUAD is another word for squared

EX: -2(squared) = -4

IMPORTANT EQUATIONS TO REMEMBER

Vertex Form

  • y= a (x-h)² + k

Factored Form

  • y= a (x-r) (x-s)

Standard Form

  • y=ax²+bx+c

Important Concepts You Should Know

Axis Of Symmetry (or AOS) - A line that divides a parabola into two congruent lines and always passes through the vertex

Zeros - Zeros are also known as the x-coordinates. (Where the graph meets the x-axis)

Optimal Value - The minimum or maximum value of a variable

QUADRATICS IN VERTEX FORM

Finite Differences 4.2

Use finite differences to determine whether each relation is linear, quadratic or neither.

When a table's first differences are the same, then the graph is considered linear.

When a table's first differences are not the same, but the second differences are the same, then the graph is considered quadratic.

Although, when neither first or second difference is the same, the graph is neither quadratic or linear.

Graphing Quadratics in Vertex Form 4.4

- y= a(x-h)²+k

Mapping Notation - (x+h, ay+k) is a strategy used to accurately graph any quadratic relation. Before using mapping notation, you need to know the key points of a basic quadratic relation (y=x²)

The basic quadratic relation is the simplest parabolic curve.

Step Pattern - (1xa, 3xa, 5xa) is when you multiply 1, 3 and 5 by the given a value to find the parabola.

Find x and y using Vertex form

To find the x intercept of a line, you simply sub in y as 0 (y=0)

To find the y intercept, you sub in x as 0 (x=0)

This allows you to isolate the x or y variable so you can solve the equation on one side of the equal sign, figuring out the x and y value.

Solving the "Ball thrown in the air" problem

Quadratics in Factored Form

Multiplying binomials and Special Products 5.1 + 5.2

REMEMBER F.O.I.L (First Outside Inside Last)


Ex.1 (h+9)(h-5) → Binomial

Expand and Simplify.

=h²+5h+9h-45

=h²+4h-45 → Trinomial


Ex.2 (3y-2)(4y+1)

Expand and Simplify.

=12y²+3y-8y-2

=12y²-5y-2


Ex.3 - MULTIPLY BINOMIALS FIRST.

-2 (x-5)(x+6)

-2 (x²+6x-5x-30)

-2 (t²+t-30)

Now bring outside number into brackets.

-2t²-2t+60


Special Cases


  • When an equation in brackets is squared, you double the equation and then simplify. Ex. (x+4)² = (x+4) (x+4)
Perfect Square - Squaring a binomial (example above)


(x+3)(x+3)
=x²-3x+3x-9 (Automatically remove the -3x and +3x.)
=x²-9

Difference of Squares - (a+b)(a-b) = a² - b²


Common Factoring 5.3


  • Opposite of expanding
Ex.1 4x + 6


Step One: Find GFC (What is the greatest number in common with 4 and 6?)

GCF = 2.

Step Two: Write solution with brackets

= 2 (2x+3)


Ex.2 12x³ - 6x²

GCF = 6x²

=6x² (2x-1)


  • To check your answer, expand and simplify the overall equation to see if you get the original equation
  • =6x² (2x-1)
  • =12x³-6x²


Factoring Simple Trinomials 5.4

Ex.1 x²+7x+12 (standard form) = (x+3)(x+4) (factored form)

When we are factoring a polynomial with the form ax²+bx+c and a=1, we want to find


  • 2 numbers that add to give b
  • 2 numbers that multiply to give c
Therefore, 3x4=12 and 3+4=7.


Factoring Complex Trinomials 5.5

Ex.1 2x²-3x-35

Step one: add the lead number (2) with the last number (-35) to get -70.

Step two: Figure out 2 numbers that add up to -3, and that multiply to get -70.

Therefore, -10x7=-70 and -10+7=-3


Factoring Differences of Squares and Perfect Square Trinomials 5.6

Differences of Squares

How would you factor 9x²-16?

Remember that (a+b)(a-b)=a²-b²

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

a=√9x²=3x

b=√16=4

9x²-16

=(3x+4)(3x-4)

Perfect Squares

(k+7)²

(k+7)(k+7)

=k²+14k+49


Solving Quadratics by Factoring (finding the zeros) 6.2

Ex. 1

Suppose you have an equation: x²+5x+6

This can be factored into (x+2)(x+3)=0


  • 2x3=6
  • 2+3=5


Then, we take the number inside the bracket and make it the opposite sign.


  • x=-2
  • x=-3
If quadratic equation cannot be factored, then the method is "Not Possible".


Graphing Quadratics in Factored Form 4.5 & 6.3


  • y=a(x-r)(x-s)

Say you're given the equation -x² - 2x + 8 = -1 (x²+2x-8)
Factor the expression in the parenthesis. Identify 2 numbers that sum up 2 and multiply to get –8. The numbers are 4 and -2

You now get the equation in factored form: y=-(x+4)(x-2)

Therefore, the x intercepts of the graph are -4 and 2.

Now, the x coordinate of the vertex will be -4+2 ÷ 2 = -1.

To find the y coordinate, you simply sub in x=-1 in the equation y=-x²-2x+8

y=-(-1)²-2(-1)=8.

=-1+2+8

=9.

Therefore, the vertex is (-1,9)

Quadratic Equations - Solving Word problems by Factoring 1c

Quadratics in Standard Form

Maximum and Minimum Values 6.1


  • Turning standard form to vertex form

Standard form - y=ax²+bx+c

Vertex form - y=a(x-h)²+k

Step One: Group x² x terms together

Ex. y=x²+8x+5

=(x²+8x)+5

Step Two: Complete the square inside the brackets

Ex. y=(x²+8x+16-16)+5


  • Get 16 by dividing 8 by 2² (=4²)
y=(x²+8x+16)-16+5


y=(x²+8x+16) is a perfect square trinomial

Step Three: Write the trinomial as a binomial squared

y=(x+4)²-16+5

y=(x+4)²-11

Vertex: (-4,11)


The Quadratic Formula 6.4

The Quadratic Formula is the solution of the quadratic equation. Other ways to solve a quadratic relation is by completing the square, factoring or graphing.

  • When you cannot factor to get the x intercepts, you simply just use the quadratic formula which is -b+/-√b²-4ac over 2a
Ex. 3x²+9x-2=0

a=3

b=9

c=-2

Sub the given coefficients, solve the equation to get the x intercepts.

Maximizing revenue-quadratic word problem

Reflection

During the start of the quadratics unit, I was having trouble with understanding the basic steps, quadratics 2 was a little bit more challenging but I knew if I could get the hang of it that quadratics 3 would be a little less difficult for me. After school, I would do homework questions and review the product so I would hopefully get a better understanding, and I thought that I did but my tests showed otherwise. On unit tests, I would usually blank out which was not beneficial for my grades, I needed a new strategy. Quadratics may not be my biggest strength but I sure did learn a lot from this unit, and the unit being broken down into separate chapters made it easier for me to understand.

In quadratics 1, I learned about parabola's, how to read them and how to plot them. Finding all the points to the parabola (vertex, axis of symmetry, x and y intercepts) took a lot of steps but once I got the hang of it, it got easier. The most easiest thing I learned was what it meant when a parabola opened upwards compared to downwards.

Quadratics 2 was slightly easier now that I sort of got the hang of quadratics 1. I learned about quadratics relations and equations like vertex form, standard form and factored form.

Finally, in quadratics 3 I was taught how to solve quadratic equations in all 3 different forms.

Quadratics is a tough unit to fully learn but once I got the hang of each chapter, I understood it more than before.