Question Video: Identifying the Graph of a Trigonometric Function Mathematics • First Year of Secondary School

Video Transcript

Identify the graph of the function 𝑓 of 𝑥 equal to two times the cos of 𝑥.

We have been given five different graphs to choose from. To answer this question, let’s use our knowledge of the properties of the general cosine function.

We’ll begin by focusing on the property which states that the 𝑦-intercept of the cosine function is one. In particular, this property means that cos of zero equals one. This property is very important for distinguishing cosine graphs from sine graphs, since sin of zero equals zero, meaning the 𝑦-intercept is different.

Let’s now consider another property of the cosine function. The cosine function is periodic, with a period of 360 degrees, or two 𝜋 radians. Recall that a periodic function repeats itself in cycles. And the period of a function is how long it takes to complete one full cycle and return to its original position. Now, sine graphs also have a period of 360 degrees. So this property does not help us distinguish between sine and cosine graphs.

Let’s consider a third useful property of the cosine function. The maximum value of the function is one, and the minimum value is negative one. This is also true for the sine function. When we consider option (A), we notice that the plot neither has a periodic shape nor a maximum or minimum value. So this cannot be the graph of 𝑦 equals two times the cos of 𝑥.

Next, we note that both graphs (D) and (E) have a 𝑦-intercept of zero. This leads us to believe these are sine functions. The remaining options have different 𝑦-intercepts. So we can distinguish between them by evaluating the 𝑦-intercept of 𝑓 of 𝑥. To do this, we will evaluate 𝑓 of zero. We know that cos of zero is one. So two times cos of zero is two. This means we are looking for a periodic graph with a 𝑦-intercept of two. Graph (C) has a 𝑦-intercept of four, whereas graph (B) has the correct 𝑦-intercept of two.

We also notice that graph (B) has a maximum value of two and a minimum value of negative two, which is in line with what we expect when the cosine function is multiplied by a constant of two. So, by comparing each graph to the key properties of the cosine function and calculating the 𝑦-intercept, we have shown that graph (B) represents the function 𝑓 of 𝑥 equals two times the cos of 𝑥.