The equation y=30((1)/(2))^(x) is graphed in the xy-plane. Which of the following statements about the graph is true? Choose 1 answer: (A) As x increases, y increases at an increasing rate. (B) As x increases, y increases at a decreasing rate. (C) As x increases, y decreases at an increasing rate. (D) As x increases, y decreases at a decreasing rate.

Question

The equation  y=30((1)/(2))^(x) is graphed in the  xy-plane. Which of the following statements about the graph is true? Choose 1 answer: (A) As  x increases,  y increases at an increasing rate. (B) As  x increases,  y increases at a decreasing rate. (C) As  x increases,  y decreases at an increasing rate. (D) As  x increases,  y decreases at a decreasing rate.

Bytelearn · Accepted Answer

Given Equation: We are given the equation y = 30(1/2)^x and we need to determine how y changes as x increases. The base of the exponent, 1/2, is less than 1, which means that as x increases, the value of (1/2)^x decreases. This is because any number between 0 and 1 raised to a higher power will get smaller.Relationship between y and x: Since y is directly proportional to (1/2)^x, as (1/2)^x decreases, y also decreases. Therefore, as x increases, y decreases. This eliminates options (A) and (B) which suggest that y increases as x increases.Rate of decrease of y: Next, we need to determine the rate at which y decreases. The function y = 30(1/2)^x is an exponential decay function. In an exponential decay, as x increases, the rate of decrease of y slows down. This is because each additional increase in x results in a smaller absolute decrease in (1/2)^x, and thus a smaller absolute decrease in y.Conclusion: Therefore, the correct statement about the graph is that as x increases, y decreases at a decreasing rate. This corresponds to option (D).

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