Which of the following graphs represents the function 𝑦 equals negative four times two to the power 𝑥?
We can see that we are given four different graphs. And we need to check which of these graphs represents this exponential function. There are two different ways in which we can approach this question. The first way might be to consider what the function 𝑦 equals two to the power of 𝑥 would look like. We could draw a table of values and then select a few different 𝑥-values and work out the corresponding 𝑦-values. So when 𝑥 is equal to zero, 𝑦 is equal to two to the power of zero. And since any value to the power of zero is one, then we have a 𝑦-value of one. When 𝑥 is equal to one, the 𝑦-value is two to the power of one, which is simply two. When the 𝑥-value is two, the 𝑦-value is four. And when the 𝑥-value is three, the 𝑦-value is eight.
We could then sketch a quick graph of the curve going through the points zero, one; one, two; two, four; and three, eight. And then we need to consider how the function 𝑦 equals two to the power 𝑥 is different to the function 𝑦 equals negative four times two to the power 𝑥. We can remember that a function 𝑓 of 𝑥 equals 𝑐 times 𝑏 to the power 𝑥 is a vertical stretch of the function 𝑓 of 𝑥 equals 𝑏 to the power 𝑥 by a scale factor of 𝑐. We can also note that this graph of 𝑓 of 𝑥 equals 𝑐 times 𝑏 to the power of 𝑥 will pass through the point zero, 𝑐. Furthermore, in this function, we have a 𝑐-value of negative four, which is negative. And so there’s also another transformation that occurs.
We can say that when 𝑐 is less than zero, the function 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥 is reflected in the horizontal axis and then stretched vertically by a factor of 𝑐 units. Let’s see what this function 𝑦 equals two to the power of 𝑥 would look like after just the reflection in the horizontal axis. So here’s the function 𝑦 equals negative two to the power 𝑥. We can see, for example, that the coordinate zero, one has been reflected to the coordinate zero, negative one. We still need to perform this vertical stretch. And here we have the function 𝑦 equals negative four times two to the power 𝑥. Notice that the 𝑦-intercept is at zero, negative four. Therefore, which of the four graphs (A), (B), (C), or (D) indicate the same shape and features of the graph we have sketched?
Well, although both graphs (C) and (D) have the same overall shape, it is only option (D) which has the same 𝑦-intercept of zero, negative four. We could then give the answer that graph (D) represents this function. The alternative method to working out the answer is by using the available answer options. We would work out the 𝑥- and 𝑦-values of some coordinates in the given function and then look at the different functions represented to see which also pass through these same coordinates. Let’s take the 𝑥-values of zero, one, and two. When 𝑥 is equal to zero, the 𝑦-value is equal to negative four times two to the power zero. This simplifies to negative four times one, and so 𝑦 is equal to negative four. The coordinate zero, negative four will lie on the function.
At this point, we can eliminate answer option (A) and answer option (C) as the only two functions here which have the coordinate zero, negative four are graphs (B) and (D). We’ll need to work out at least one more coordinate to determine which graph is correct. So when 𝑥 is equal to one, we have a 𝑦-value which is negative four times two to the power one. That’s negative four times two, which gives us a value of negative eight. The coordinate one, negative eight is in the graph in option (D). However, it is not in option (B). We don’t need to work out any further coordinate values in order to give the answer that it is option (D) which represents the function 𝑦 equals negative four times two to the power 𝑥.