identify the graph of $y = x^2$.

Explanation:

Step1: Recall the properties of ( y = x^2 )

The function ( y = x^2 ) is a quadratic function. Its graph is a parabola. For a quadratic function in the form ( y = ax^2 + bx + c ) (here ( a = 1 ), ( b = 0 ), ( c = 0 )), when ( a>0 ), the parabola opens upwards. Also, the vertex of ( y = x^2 ) is at the origin ((0,0)) because when ( x = 0 ), ( y = 0 ), and the axis of symmetry is the ( y )-axis (( x = 0 )).

Step2: Analyze the given graphs

• The first graph: The parabola opens downwards (since the arms of the parabola point down). For ( y = x^2 ), ( a = 1>0 ), so it should open upwards. So this is not the graph of ( y = x^2 ).
• The second graph: The parabola opens upwards, has its vertex at the origin ((0,0)), and is symmetric about the ( y )-axis. This matches the properties of ( y = x^2 ).
• The third graph: This is not a parabola (it looks like a horizontal ellipse or a different conic section), and the graph of ( y = x^2 ) is a parabola, so this is not the graph of ( y = x^2 ).

Answer:

The Middle Graph (the second graph among the three given graphs)