Video: Identifying Graphs of Exponential Equations

Which of the following graphs represents the equation 𝑦 = (1/4)^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 a quarter to the power of π‘₯.

It’s useful to begin by spotting that this is an exponential equation. An exponential equation is one of the form 𝑦 equals 𝑏 to the power of π‘₯, where 𝑏 is a real positive constant not equal to one. Now, we know several things about the graphs of exponential equations. We know that their 𝑦-intercepts for a start are one. They pass through the point zero, one. And so we can instantly eliminate three of our graphs. We can eliminate A, B, and C. Graph A actually intersects at zero as does graph C, whereas graph B doesn’t appear to intersect the 𝑦-axis at all.

Now, we also know something about the shape of these curves. If our value for 𝑏 is greater than one, then we’re representing exponential growth. And the graph looks a little something like this. Notice that the π‘₯-axis represents a horizontal asymptote of our graph. It gets closer and closer but never quite touches it. Now, if 𝑏 is greater than zero and less than one, we have exponential decay. Our graph is decreasing over its entire domain. The π‘₯-axis is still a horizontal asymptote to our graph, but this time it looks a little like this.

So what’s our value of 𝑏? Well, the equation is 𝑦 equals a quarter to the power of π‘₯. So 𝑏 is equal to a quarter which is greater than zero and less than one. That tells us that our graph represents exponential decay. It’s going to be decreasing over its entire domain. We can see that that’s graph D.

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