Question Video: Identifying Graphs of Exponential Equations Mathematics

Which of the following graphs represents the equation 𝑦 = 3^π‘₯? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

Which of the following graphs represents the equation 𝑦 equals three to the power of π‘₯?

Our equation 𝑦 equals three to the power of π‘₯ represents an exponential equation. So let’s recall what we know about exponential functions. Firstly, we know that an exponential function of the form 𝑓 of π‘₯ equals 𝑏 to the power of π‘₯, where 𝑏 is a positive real constant, passes through at zero, one. In other words, it passes through the 𝑦-axis at one. So let’s see if we can eliminate any of our graphs from our question. Graph B passes through at zero. Graph C doesn’t seem to intersect the 𝑦-axis at all. Graph E intersects at negative one. And that leaves us with Graph A and Graph D, which both intersect the 𝑦-axis at one.

There’s two ways that we can check which one of our graphs is correct. We could pick a point and test this. For example, our first graph passes through the point with coordinates one, three. Let’s let π‘₯ be equal to one, since the π‘₯-coordinate is one, and see if the 𝑦-coordinate is indeed three. If π‘₯ is equal to one, 𝑦 is equal to three to the power of one, which is indeed three. And so we can infer that the graph of the equation 𝑦 equals three to the power of π‘₯ must pass through the point one, three. And so our graph is A.

There is, however, another way we could have tested this. We know that if 𝑏 is greater than one, our graph represents exponential growth. In other words, it’s always increasing. Whereas if 𝑏 is between zero and one, it represents exponential decay; it’s always decreasing. We can see that the graph of D is decreasing over its entire domain. It’s always sloping downwards. And so the value of 𝑏, the base if you will, needs to be between zero and one. So this could be 𝑦 equals one-third to the power of π‘₯, for example. The correct answer here then is A.

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