Learning About Quadratics
Fun with Quadratics!
What is a Quadratic?
QUAD is another word for squared
EX: -2(squared) = -4
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IMPORTANT EQUATIONS TO REMEMBER
- y= a (x-h)² + k
- y= a (x-r) (x-s)
Important Concepts You Should Know
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
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.
- 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.
Quadratics in Factored Form
REMEMBER F.O.I.L (First Outside Inside Last)
Ex.1 (h+9)(h-5) → Binomial
Expand and Simplify.
=h²+4h-45 → Trinomial
Expand and Simplify.
Ex.3 - MULTIPLY BINOMIALS FIRST.
Now bring outside number into brackets.
- When an equation in brackets is squared, you double the equation and then simplify. Ex. (x+4)² = (x+4) (x+4)
=x²-3x+3x-9 (Automatically remove the -3x and +3x.)
Difference of Squares - (a+b)(a-b) = a² - b²
Common Factoring 5.3
- Opposite of expanding
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
- To check your answer, expand and simplify the overall equation to see if you get the original equation
- =6x² (2x-1)
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
Factoring Complex Trinomials 5.5
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.
Solving Quadratics by Factoring (finding the zeros) 6.2
Suppose you have an equation: x²+5x+6
This can be factored into (x+2)(x+3)=0
Then, we take the number inside the bracket and make it the opposite sign.
Graphing Quadratics in Factored Form 4.5 & 6.3
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
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
Therefore, the vertex is (-1,9)
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
Step Two: Complete the square inside the brackets
- Get 16 by dividing 8 by 2² (=4²)
y=(x²+8x+16) is a perfect square trinomial
Step Three: Write the trinomial as a binomial squared
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
Sub the given coefficients, solve the equation to get the x intercepts.
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.