Parabola

Finding the Vertex of a Parabola ,[object Object]

Example 1: ,[object Object],Completing the Square if the coefficient of x 2 is 1.

Example 1: ,[object Object],Completing the Square if the coefficient of x 2 is 1. f(x)=(x 2 -4x )+3 Separate the x-terms

Example 1: Continued ,[object Object],f(x)=(x 2 -4 x )+3 Square half of the coefficient of x ( ) -4 2 2 =4 __

Example 1: Continued ,[object Object],-4 2 f(x)=(x 2 -4x )+3 ( ) 2 = 4 f(x)=(x 2 -4x + 4 )+3 -4 Add this constant inside the parentheses, and subtract it on the outside

Example 1: Continued ,[object Object],f(x)=(x 2 -4x + 4)+3-4

Example 1: Continued ,[object Object],f(x)=( x 2 -4x + 4 )+3-4 The expression in the parentheses is a perfect square trinomial

Example 1: Continued ,[object Object],f(x)=( x 2 -4x + 4 )+3-4 Factor it! f(x)=( x-2 )( x -2 )+3-4

Example 1: Continued ,[object Object],f(x)=(x 2 -4x + 4)+3-4 Simplify the right side f(x)=(x-2)(x -2)+ 3-4 f(x)=(x-2) 2 -1

Example 1: Continued ,[object Object],f(x)=(x-h) 2 +k where (h,k) is the vertex f(x)=(x-2) 2 -1

Example 1: Continued ,[object Object],Subtraction is like adding the opposite f(x)=(x-2) 2 + ( - 1) f(x)=(x-h) 2 +k where (h,k) is the vertex f(x)=(x-2) 2 - 1

Example 1: Continued ,[object Object],f(x)=(x- 2 ) 2 +( -1 ) f(x)=(x- h ) 2 + k where (h,k) is the vertex f(x)=(x-2) 2 -1 For our function, h=2 and k=-1

Example 1: Continued ,[object Object],f(x)=(x-2) 2 +(-1) f(x)=(x-h) 2 +k where ( h , k ) is the vertex f(x)=(x-2) 2 -1 For our function, h=2 and k=-1 Therefore, the vertex is ( 2 , -1 )

Example 1: Completed Here is the graph of f(x)=x 2 -4x+3 Vertex is (2, -1)

Example 2: ,[object Object],Completing the Square if the coefficient of x 2 is not 1.

Example 2: Continued ,[object Object],f(x)=(-2x 2 -2 x )+1 Separate the x-terms

Example 2: Continued ,[object Object],f(x)=( -2 x 2 -2 x )+1 Factor out the coefficient of x 2 f(x)= -2 (x 2 + x )+1

Example 2: Continued ,[object Object],f(x)=(-2x 2 -2 x )+1 f(x)= -2 (x 2 + 1 x )+1 Square half of the coefficient of x ( ) 2 = __ 2 __ 4 1 1

Example 2: Continued ,[object Object],__ f(x)=(-2x 2 -2 x )+1 f(x)=-2(x 2 +1x )+1 ( ) 1 2 = 2 f(x)= -2(x 2 +x+ )+1 Add this constant inside the parentheses __ 4 1 __ 4 1

Example 2: Continued ,[object Object],__ f(x)=(-2x 2 -2 x )+1 f(x)=-2(x 2 +1x )+1 ( ) 1 2 = 2 f(x)= -2 (x 2 + x+ )+1 Notice we have really added -2 ( ) to the equation __ 4 1 __ 4 1 __ 4 1

Example 2: Continued ,[object Object],__ f(x)=(-2x 2 -2 x )+1 f(x)=-2(x 2 +1x )+1 ( ) 1 2 = 2 f(x)= -2 (x 2 + x+ )+1 -( -2 )( ) Therefore, subtract -2 ( ) to maintain the same equation __ 4 1 __ 4 1 __ 4 1 __ 4 1

Example 2: Continued ,[object Object],__ f(x)=(-2x 2 -2 x )+1 f(x)=-2(x 2 +1x )+1 ( ) 1 2 = 2 f(x)= -2(x 2 +x+ )+ 1-(-2)( ) Simplify f(x)= -2(x 2 +x+ )+ __ 4 1 __ 4 1 __ 4 1 __ 4 1 __ 2 3

Example 2: Continued ,[object Object],3 f(x)= -2(x 2 +x+ )+ __ 4 1 __ 2

Example 2: Continued ,[object Object],3 f(x)= -2( x 2 +x+ )+ __ 4 1 __ 2 The expression in the parentheses is a perfect square trinomial

Example 2: Continued ,[object Object],3 f(x)= -2( x+ )( x + )+ f(x)= -2( x 2 +x+ )+ __ 4 1 __ 2 Factor it! __ 2 1 __ 2 1 __ 2 3

Example 2: Continued ,[object Object],3 Simplify the right side f(x)= -2( x+ )( x + )+ f(x)=-2 ( x+ ) 2 + f(x)= -2(x 2 +x+ )+ __ 4 1 __ 2 __ 2 1 __ 2 1 __ 2 3 __ 2 3 __ 2 1

Example 2: Continued ,[object Object],f(x)=a(x-h) 2 +k where (h,k) is the vertex f(x)=-2 (x+ ) 2 + __ 2 3 __ 2 1

Example 2: Continued ,[object Object],f(x)=a(x-h) 2 +k where (h,k) is the vertex f(x)=-2 (x - ) 2 + Change addition to subtracting the opposite f(x)=-2 (x + ) 2 + __ 2 3 __ 2 - 1 __ 2 3 __ 2 1

Example 2: Continued ,[object Object],f(x)=a(x- h ) 2 + k where (h,k) is the vertex f(x)=-2 (x- ) 2 + For our function, h= and k= f(x)=-2 (x+ ) 2 + __ 2 3 __ 2 -1 __ 2 3 __ 2 -1 __ 2 3 __ 2 1

Example 2: Continued ,[object Object],f(x)=a(x-h) 2 +k where ( h , k ) is the vertex f(x)=-2 (x- ) 2 + For our function, h= and k= f(x)=-2 (x+ ) 2 + Therefore, the vertex is ( , ) __ 2 3 __ 2 -1 __ 2 3 __ 2 -1 __ 2 3 __ 2 1 __ 2 -1 __ 2 3

Example 2: Completed Here is the graph of f(x)= -2x 2 -2x+1 Vertex is ( , ) __ 2 -1 __ 2 3

Parabola

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Parabola