Questions: The graph shows a parabola and its directrix. Write the equation of the parabola in vertex form.

Transcript text: The graph shows a parabola and its directrix. Write the equation of the parabola in vertex form.

Solution

Solution Steps

Step 1: Identify the Vertex

The vertex of the parabola is the lowest point on the graph. From the graph, the vertex is at the origin (0, 0).

Step 2: Determine the Direction of the Parabola

The parabola opens upwards, as indicated by the shape of the curve.

Step 3: Identify the Directrix

The directrix is a horizontal line below the vertex. From the graph, the directrix is at \( y = -2 \).

Step 4: Calculate the Distance from the Vertex to the Directrix

The distance from the vertex (0, 0) to the directrix \( y = -2 \) is 2 units.

Step 5: Write the Equation in Vertex Form

The vertex form of a parabola that opens upwards is \( y = a(x - h)^2 + k \). Here, \( (h, k) \) is the vertex, and \( a \) is a constant. Since the vertex is at (0, 0), the equation simplifies to \( y = ax^2 \).

Step 6: Determine the Value of \( a \)

The value of \( a \) is given by \( \frac{1}{4p} \), where \( p \) is the distance from the vertex to the directrix. Here, \( p = 2 \), so \( a = \frac{1}{4 \times 2} = \frac{1}{8} \).