Video Transcript
Find the range of the function 𝑓 of 𝜃 is equal to eight sin of seven 𝜃.
Our function is eight sin of seven 𝜃. And so to answer this, we’re going to begin by recalling what we know about the function 𝑓 of 𝜃 is equal to sin 𝜃. And then the transformations that map it onto eight sin of seven 𝜃.
The graph of 𝑓 of 𝜃 is equal to sin 𝜃 is periodic. It repeats every 360 degrees. It has a maximum at one and a minimum of negative one. And then we recall how to transform functions. For 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑎 times 𝑥 is a horizontal stretch by a scale factor of one over 𝑎. 𝑦 is equal to 𝑏 times 𝑓 of 𝑥 is a vertical stretch. This time, the scale factor is 𝑏.
Well, our function is eight sin of seven 𝜃. So, 𝑎 must be equal to seven and 𝑏 must be equal to eight. We’re going to stretch our function vertically by a scale factor of eight and horizontally by a scale factor of one-seventh. Now, this isn’t to scale, but this will give us something like the pink plot shown.
Now, we want to find the range of this function. The domain is the input, and the range is the output we get when we input that domain. So, what’s the output of our function? Well, we see that the output is as high as eight and as low as negative eight. This means eight sin seven 𝜃 will have values in the closed interval negative eight to eight. And this is the range of our function. It’s the closed interval from negative eight to eight.
