Question

For what intervals is ( y = 5 - x^2 ) positive? For what interval is the function negative?

a) Positive: ( x > -sqrt{5} ); Negative: ( x < -sqrt{5} )
b) Positive: ( x < sqrt{5} ); Negative: ( x > sqrt{5} )
c) Positive: ( x < -sqrt{5} ); Negative: ( x > sqrt{5} )
d) Positive: ( x > sqrt{5} ); Negative: ( x < sqrt{5} )

Answer 1

The function y = 5 - x^2 is positive for values of x between -sqrt(5) and sqrt(5), and negative for values of x less than -sqrt(5) and greater than sqrt(5). The correct intervals are given in option (b).

Step-by-step explanation:

The student is asking for the intervals where the function y = 5 - x^2 is positive or negative. To find these intervals, let's analyze the function and its graph. The graph of y = -x^2 is a downward-opening parabola with vertex at (0, 0). Therefore, the graph of y = 5 - x^2 will also be a downward-opening parabola, but with its vertex shifted to (0, 5).

To determine where the function is positive, we look for values of x where y > 0, which is above the x-axis. The function is positive between the x-intercepts where the parabola crosses the x-axis. Set y to zero and solve for x: 0 = 5 - x^2, giving us x = ±sqrt{5}. Therefore, the function is positive between -sqrt{5} and sqrt{5} (x is between -sqrt{5} and sqrt{5}).

For where the function is negative, you consider the values outside of these x-intercepts. The function is negative for x < -sqrt{5} and x > sqrt{5}. By examining the intervals given in the options, option (b) matches our findings: Positive for x < sqrt{5}, and Negative for x > sqrt{5}.