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Quadratic Functions By: Rebekah, Tara, Sam & Mel

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

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

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

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

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

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

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

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

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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 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 OR 10 < x because the second circle is an open circle

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

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

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

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

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

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

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

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

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3. Neither

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

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

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y=x 2 y=-x 2 y= ½ x 2 Characteristics of Quadratic Function

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

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

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

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

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