Questions: Enter the rule of the function.

Solution Steps

The graph shows a piecewise function with two segments:

1. A quadratic segment from \( x = -4 \) to \( x = 0 \).
2. Another quadratic segment from \( x = 0 \) to \( x = 5 \).

For the first segment:

• The points are \((-4, 16)\), \((0, 0)\).
• The general form of a quadratic function is \( y = ax^2 + bx + c \).
• Since the vertex is at \((0, 0)\), the equation simplifies to \( y = ax^2 \).
• Using the point \((-4, 16)\): \[ 16 = a(-4)^2 \implies 16 = 16a \implies a = 1 \]
• Therefore, the equation for the first segment is \( y = x^2 \).

For the second segment:

• The points are \((0, 0)\), \((5, 25)\).
• The general form of a quadratic function is \( y = ax^2 + bx + c \).
• Since the vertex is at \((0, 0)\), the equation simplifies to \( y = ax^2 \).
• Using the point \((5, 25)\): \[ 25 = a(5)^2 \implies 25 = 25a \implies a = 1 \]
• Therefore, the equation for the second segment is \( y = x^2 \).

Combine the equations for the two segments into a piecewise function: \[ f(x) = \begin{cases} x^2 & \text{if } -4 \leq x \leq 0 \\ x^2 & \text{if } 0 \leq x \leq 5 \end{cases} \]

Final Answer