### Video Transcript

Which of the following could be the equation of the curve? Option (A) π¦ equals negative four times one plus three to the power π₯. Option (B) π¦ equals four times one plus three to the power π₯. Option (C) π¦ equals four times one minus three to the power negative π₯. Option (D) π¦ equals four times one plus three to the power negative π₯. Or (E) π¦ equals four times one minus three to the power π₯.

So letβs take a look at this given curve, which is in blue on the graph. We might notice that this curve passes through the coordinate zero, zero. This means that in the equation of the curve, when π₯ is equal to zero, π¦ is equal to zero. We can use the answer options to test whether or not this is the case in each of the possible equations. Starting with answer option (A), when π₯ is equal to zero, the right-hand side of this equation is negative four times one plus three to the power zero. We can simplify this as negative four times one plus one. Negative four times two is equal to get negative eight. And thatβs not equal to zero.

In this equation, when π₯ is equal to zero, π¦ is not equal to zero. And so we can eliminate answer option (A). In the same way, substituting π₯ equals zero into the right-hand side of the equation in option (B), we would have four times one plus three to the power zero, but four times two is equal to eight. And thatβs also not equal to zero. So we can eliminate answer option (B).

Substituting π₯ equals zero into equation (C), we have four times one minus three to the power of negative zero. And three to the power negative zero is equivalent to three to the power of zero. When we simplify this, we get four times one minus one, and four times zero is equal to zero. And so in this equation given in answer option (C), when π₯ is equal to zero, π¦ is equal to zero. So the equation in option (C) is still a possible answer.

Letβs continue the same process for options (D) and (E). So substituting π₯ equals zero into answer option (D), we get a π¦-value that is not equal to zero. And so we can eliminate answer option (D). However, in answer option (E), when π₯ is equal to zero, π¦ is equal to zero. We have now eliminated three of the answer options. So in order to determine which of answer options (C) or (E) is the correct one, letβs consider the general behavior of the graph.

We can see that for the positive π₯-values, the π¦-values very quickly become extremely large and negative. Therefore, we could try substituting a positive π₯-value, for example, π₯ equals one, into each of the remaining options. So in equation (C), when we substitute π₯ equals one into the right-hand side, we would have four times one minus three to the power of negative one.

We can recall that three to the power of negative one is equivalent to one-third. Then, one minus one-third is simplified to two-thirds. Four multiplied by two-thirds is equal to eight over three or eight-thirds. In the same way, when we substitute π₯ equals one into the right-hand side of the equation in option (E), we get four times one minus three to the power of one. This simplifies to negative eight.

So remember what we said about the behavior of this graph. For positive π₯-values, the π¦-values become very quickly extremely large and negative. When π₯ is greater than zero, weβll have a π¦-value, which is less than zero. For the equation given in option (C), the π₯-value that we chose was positive, but the π¦-value was also positive. In equation (E), when π₯ was positive, π¦ was negative. We can therefore eliminate answer option (C), which means that the equation of the curve is that given in option (E). Itβs π¦ equals four times one minus three to the power of π₯.