Question

The graph of a cosine function is drawn. One full cycle goes from x = 0 to x = 2 and the low point on that cycle is (1, — 5).Which of the following functions could have this graph?○ y = 5.cosmxy = 5. cos2xOy=-5 cos xxO y=-5-cos2x

Answer 1

y=5 cos πx

Explanation:

In the graph of the cosine function:

One full cycle goes from x = 0 to x = 2, this means that the period of the graph is 2.

For the general cosine function:

`f(x)=a\cos(bx+c)+d`

The period is determined using the formula:

`Period=(2\pi)/(b)`

Thus, to have a period of 2, the value of b must be π.

The options that satisfy these properties are:

`\begin{gathered} y=5\cos\pi x \\ y=-5\cos\pi x \end{gathered}`

Next, the low point on that cycle is (1, — 5).

`\begin{gathered} \text{ When }x=1:y=5\cos\pi x=5\cos\pi(1)=-5 \\ \text{When }x=1:y=-5\cos\pi x=-5\cos\pi(1)=5 \end{gathered}`

The function of these two that contain (1, -5) is y=5 cos πx.

The function that could have this graph is y=5 cos πx.