which of the following describes the function shown in the table below?\n| x | y |\n| -4 | 16 |\n| -1 | 2…

Explanation:

Step1: Analyze the pattern of y - values

Let's check the ratio between consecutive y - values. For (x=-4,y = 16) and (x=-1,y = 2), the ratio is (\frac{2}{16}=\frac{1}{8})? Wait, no, the change in (x) from (-4) to (-1) is (3) units. Wait, maybe we should check the ratio when (x) increases by a certain factor or when we look at the multiplicative relationship. Let's check the ratio between (y) - values when (x) changes. From (x=-4,y = 16) to (x=-1,y = 2): The change in (x) is (\Delta x=-1-\left(-4\right)=3). The ratio of (y) - values is (\frac{2}{16}=\frac{1}{8}). From (x=-1,y = 2) to (x = 2,y=0.25): (\Delta x=2-\left(-1\right)=3). The ratio of (y) - values is (\frac{0.25}{2}=\frac{1}{8}). From (x = 2,y = 0.25) to (x = 4,y=0.0625): (\Delta x=4 - 2=2). The ratio of (y) - values is (\frac{0.0625}{0.25}=\frac{1}{4})? Wait, maybe a better way is to see the multiplicative factor. Let's assume the function is of the form (y = ab^{x}). Let's take two points. Let's take ((x_1,y_1)=(-4,16)) and ((x_2,y_2)=(-1,2)). We have (16=ab^{-4}) and (2=ab^{-1}). Divide the second equation by the first equation: (\frac{2}{16}=\frac{ab^{-1}}{ab^{-4}}). Simplify the right - hand side: (\frac{ab^{-1}}{ab^{-4}}=b^{-1-\left(-4\right)}=b^{3}). So (\frac{1}{8}=b^{3}), then (b=\frac{1}{2}) (since (\left(\frac{1}{2}\right)^{3}=\frac{1}{8})). Now, substitute (b = \frac{1}{2}) into (2=ab^{-1}). We know that (b^{-1}=2), so (2=a\times2), then (a = 1). So the function is (y=\left(\frac{1}{2}\right)^{x}). As (x) increases, (y) - values decrease. For example, when (x=-4), (y = 16); when (x=-1), (y = 2); when (x = 2), (y=0.25); when (x = 4), (y = 0.0625); when (x = 5), (y=0.03125). The (y) - values are getting smaller as (x) increases, which means it's an exponential decay function. A quadratic function has a constant second - difference (the difference of differences of (y) - values). Let's check the first differences of (y) - values: From (x=-4) to (x=-1): (2 - 16=-14) From (x=-1) to (x = 2): (0.25 - 2=-1.75) From (x = 2) to (x = 4): (0.0625 - 0.25=-0.1875) From (x = 4) to (x = 5): (0.03125 - 0.0625=-0.03125) The first differences are not constant, and the second differences (difference of first differences) will also not be constant, so it's not a quadratic function. Also, since the (y) - values are decreasing as (x) increases, it's not a growth function. So the function is exponential with a continual rate of decay or decrease.

Answer:

exponential, there is a continual rate of decay or decrease