Determine the vertex form of g(x) = x^2 + 2x – 1. Which graph represents g(x)?

To convert the quadratic function g ( x ) = x 2 + 2 x − 1 into vertex form, which is g ( x ) = a ( x − h ) 2 + k , we can complete the square.
Here are the steps to do this:

Start with the standard form of the quadratic:
g ( x ) = x 2 + 2 x − 1

Isolate the constant term:
g ( x ) = ( x 2 + 2 x ) − 1

Complete the square for the quadratic expression inside the parentheses:

Take the coefficient of x , which is 2, divide it by 2, and square it: ( 2 2 ​ ) 2 = 1 .

Add and subtract this square inside the parentheses:
g ( x ) = ( x 2 + 2 x + 1 − 1 ) − 1

This simplifies to:
g ( x ) = (( x + 1 ) 2 − 1 ) − 1



Simplify the expression:
g ( x ) = ( x + 1 ) 2 − 2


Now we have the vertex form g ( x ) = ( x + 1 ) 2 − 2 , where:

a = 1 (the coefficient of ( x − h ) 2 )
h = − 1 , the x-coordinate of the vertex
k = − 2 , the y-coordinate of the vertex

Therefore, the vertex of the parabola is at ( − 1 , − 2 ) .
Graphically, this means the parabola opens upward (since 0"> a > 0 ), and it is shifted 1 unit left and 2 units down from the origin.
If you plot the function g ( x ) = ( x + 1 ) 2 − 2 , the graph will represent this equation with its vertex at the point ( − 1 , − 2 ) and it will be symmetrical about the vertical line x = − 1 .

Answered by danjohnbrain | 2025-07-06