Question

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)