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

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (12)

Similar to Parabola

Similar to Parabola (20)

Recently uploaded

Recently uploaded (20)

Parabola