Questions: Find a formula for the quadratic function depicted in the following graph.

Solution Steps

The vertex of the parabola is the lowest point on the graph. From the graph, the vertex is at the point (0, -9).

Since the vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex, we can substitute \( h = 0 \) and \( k = -9 \) into the equation: \[ f(x) = a(x-0)^2 - 9 \] \[ f(x) = ax^2 - 9 \]

To find the value of 'a', we need another point on the graph. From the graph, we can see that the point (3, 0) lies on the parabola. Substitute \( x = 3 \) and \( f(x) = 0 \) into the equation: \[ 0 = a(3)^2 - 9 \] \[ 0 = 9a - 9 \] \[ 9a = 9 \] \[ a = 1 \]

Final Answer