Quadratic Functions

Quadratic Functions By: Rebekah, Tara, Sam & Mel

-3 5 Three Types of Quadratic Functions • Number Line • Set Notation { x | -3 ≤ x < 5} • Interval Notation [-3, 5)

3 9 Number Line • If the number line has a solid circle then the that means that the number is included • If the number line has an open circle then the number is not included

Set Notation { x | -6 < x ≤ 8} • “x such that -6 is less than x, which is less than or equal to 8” • < less than • ≤ less than or equal to

Interval Notation [-8, 5) • Square bracket means including: [ ] • Round bracket means not including: ( )

-3 9 Example: Use set and internal notation to describe the following • The first circle is solid so the sign after -3 is ≤ • The second circle is hollow so the sign before 9 is < Set: {x | -3 ≤ x < 9} • The first circle is solid so the bracket is a square one • The second circle is hollow so the bracket is a circular one Interval: [-3, 9)

-5 Example: Use set and internal notation to describe the following Set: {x | -5 ≤ x } • The first circle is solid so the sign after -3 is ≤ • But there isn’t a second number so it ends at X • The first circle is solid so the bracket is a square one • These isn’t a second number, and because the arrow goes on forever there is an ∞ symbol with a round bracket Interval: [-5, ∞)

8 Example: Use set and internal notation to describe the following • The first number is ∞ and the second is 8, so the ∞ is represented by X • The circle is empty so a < sign is used Set: {x | x < 8 } • The first number ∞ • The second number is 8 and it is an empty circle so the bracket is round Interval: (-∞, 8)

Example: Use set and internal notation to describe the following Set: {x | x Є R} • There are no numbers on the line so it is an element of all reals Interval: (-∞, ∞) • The first number -∞ • The second number is ∞ • This line includes every positive and negative number

-2 0 10 Example: Use set and internal notation to describe the following Set: {x | -2 ≤ x ≤ 0 or 10 < x} • The first circle is solid so the sign after -2 is ≤, and the second number is 0 • OR 10 < x because the second circle is an open circle

-2 0 10 Example: Interval: [-2, 0] (10, ∞) • The first two numbers are solid circles, you then use square brackets • Then the 10 is with an open circle, you use a round bracket. Since the arrow goes onto infinity you add a ∞

Example: Domain: {x | -4 ≤ x ≤ 2 Range: {y | -3 ≤ y ≤ 3} • This is now a horizontal and a vertical or a domain and range • When having a shape on a graph unless otherwise told its always as if these are solid circles 3 -4 2 -3 • Don’t forget that when you are dealing with range you replace x with y

Example: • You have to separate the horizontal and vertical components 2 3 -4 -2 2 -5 • Domain: {x | -4 ≤ x ≤ 2 or 3 < x} • Range: {y | -5 ≤ y ≤ -2 or 2 < y} 2 3 -4 2 -5 -2

Double Arrow Cases • There will be : • 2 Arrows up or • 2 Arrows down • For both examples the domain will always be {xlxЄR} or (-∞, ∞) • When writing the range make sure you go from the bottom of the graph to the top (negative to positive)

-5 Example: D = {xlxЄR} =(-∞, ∞) R = {yl-5≤y} = [-5, ∞)

3 Example: D = {xlxЄR} =(-∞, ∞) R = {yly ≤ 3} = (-∞, 3]

-3 Double Arrow Cases D = {xlxЄR} =(-∞, ∞) R = {yly≤3} = (-∞, -3]

Functions 1. Linear function (straight line) (y=mx+b) m= slope b=y intercept 2. Quadratic function (Parabola) y=x2

3. Neither

Always negative Completing the Square • Divide 6 by 2 and fill it in the first blank • Square 3 and place the answer in the second blank • Simplify the numbers y=x2+6x-7 y=(x+__)2-__-7 y=(x+3)2 -__-7 y=(x+3)2-9 -7 y=(x+3)2-16

Divide everything by two in order to get rid of the 2 attached to the X2 Then continue as before Completing The Square y=2x2+24x -8 2 y=2[x2+12x -4] y=2[(x+6)2-36 -4] y=2[(x+6)2 -40]

Characteristics of Quadratic Function y=x2 y=½x2 y=-x2

y=x2 - 2 y=x2 + 2 y=-(x+2)2

y=2x2 y=(x+2)2-5 Y=(x-3)2 -3

y=-(x-4)2 +3 y=-(x+2)2 y=(x-2)2 -3

y= -(x)2 -2 y= (x-4)2+3 y=(x-1)2

Quadratic Functions