Question Video: Finding the Range of a Given Sine Function Mathematics • First Year of Secondary School

Video Transcript

Find the range of the function 𝑓 of 𝜃 equals three sin 𝜃 plus four.

Let’s recall first some key properties of the sine function. First, it’s a periodic function with a period of two 𝜋 radians or 360 degrees. The graph oscillates between negative one and one, taking every value in between. This tells us that the range of the sine function is the closed interval from negative one to positive one. We can use this information to determine the range of the function 𝑓 of 𝜃.

Remember, the range is the set of all possible values of the function itself or all possible output values. So we’re looking for all possible values of the expression three sin 𝜃 plus four when 𝜃 is any real number. Let’s consider the function transformations that have been applied to sin 𝜃 to give three sin 𝜃 plus four.

Multiplying sin 𝜃 by three stretches the graph of the function vertically by a factor of three. This has the effect of multiplying each value in the range by three. So, after multiplying by three, the new range will be the closed interval from negative three to positive three.

Adding four to the function shifts the graph vertically upwards by four units. This has the effect of increasing each value in the range by four. So, in particular, negative three becomes one and three becomes seven. The range of the function 𝑓 of 𝜃 equals three sin 𝜃 plus four is therefore the closed interval from one to seven.

We could also show this by using inequalities. If sin 𝜃 is greater than or equal to negative one and less than or equal to positive one, then three sin 𝜃 is greater than or equal to negative three and less than or equal to positive three. And three sin 𝜃 plus four is greater than or equal to one and less than or equal to seven.

Using two methods then, we’ve shown that the range of the function 𝑓 of 𝜃 equals three sin 𝜃 plus four is the closed interval from one to seven.