Question Video: Identifying Graphs of Exponential Equations Mathematics

Video Transcript

Which of the following graphs represents the equation 𝑦 equals two times three to the power of 𝑥.

Now, whilst it may not look like it, this is an example of an exponential equation. It’s essentially a multiple of its general form 𝑦 equals 𝑏 to the power of 𝑥, where 𝑏 is a positive real constant not equal to one. This time, though, it’s of the form 𝑎𝑏 to the power of 𝑥. Remember, according to the order of operations, we apply the exponent before multiplying. So this is three to the power of 𝑥 times two. And this means we’re going to need to recall what we know about the transformations of graphs. Well, for a graph of the function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑥 plus some constant 𝑎 is a translation by zero 𝑎. It moves 𝑎 units up.

The graph of 𝑦 equals 𝑓 of 𝑥 plus 𝑏 is a translation by negative 𝑏 zero. This time it moves 𝑏 units to the left. Now, if we look at our equation, we see that we haven’t added a constant at all. So we recall the other rules we know. 𝑦 is equal to some constant 𝑎 times 𝑓 of 𝑥 is a vertical stretch or enlargement by a scale factor of 𝑎. Whereas 𝑦 equals 𝑓 of 𝑏𝑥 is a horizontal stretch by scale factor one over 𝑏. Now going back to our equation, we have three to the power of 𝑥. And we’re timesing the entire function by two. And so we’re looking at a vertical stretch. In fact, we need to perform a vertical stretch of the function 𝑦 equals three to the power of 𝑥 by a scale factor of two.

So what does the graph of 𝑦 equals three to the power of 𝑥 look like. It’s an exponential function, and the base is greater than one. That means our function represents exponential growth. This means we can eliminate graphs A and B. They actually represent exponential decay, since they’re decreasing; they’re sloping downwards. So we need to choose from C, D, and E. And so we also recall that the function 𝑦 equals 𝑏 to the power of 𝑥 passes through the 𝑦-axis at one. Our function 𝑦 equals three to the power of 𝑥 will do the same. It’ll pass through zero, one. But it’s been stretched vertically by a scale factor of two. This means our function 𝑦 equals two times three to the power of 𝑥 must pass through at zero, two.

Out of C, D, and E, the only function that does so is E. C passes through at one and D passes through at three. And so the graph that represents the equation 𝑦 equals two times three to the power of 𝑥 is E.