Questions: Which one of the following is the correct sign diagram for the function y = 4x - x^2?

Solution Steps

The polynomial \( y = 4x - x^2 \) can be factorized as follows: \[ y = -x(x - 4) \] This indicates that the roots of the polynomial are \( x = 0 \) and \( x = 4 \).

Next, we analyze the sign of the function \( y \) in the intervals determined by the roots \( x = 0 \) and \( x = 4 \). We will evaluate the sign of \( y \) in the following intervals: \( (-\infty, 0) \), \( (0, 4) \), and \( (4, \infty) \).

1. Interval \( (-\infty, 0) \):

   • Test point: \( x = -1 \)
   • Calculation: \[ y(-1) = 4(-1) - (-1)^2 = -4 - 1 = -5 \]
   • Sign: \( - \)

2. Interval \( (0, 4) \):

   • Test point: \( x = 2 \)
   • Calculation: \[ y(2) = 4(2) - 2^2 = 8 - 4 = 4 \]
   • Sign: \( + \)

3. Interval \( (4, \infty) \):

   • Test point: \( x = 5 \)
   • Calculation: \[ y(5) = 4(5) - 5^2 = 20 - 25 = -5 \]
   • Sign: \( - \)

Based on the sign evaluations, we can summarize the sign of the function \( y \) in each interval:

• For \( x \in (-\infty, 0) \): \( y < 0 \) (negative)
• For \( x \in (0, 4) \): \( y > 0 \) (positive)
• For \( x \in (4, \infty) \): \( y < 0 \) (negative)

Final Answer