Question

Identify the phase shift, vertical translation and range for each function.a.

Answer 1

The phase shift represents how much the function is shifting to the right or left. The vertical translation refers to how the graph of the function is shifted up or down. The range of a function is the set of all possible y-values the function can take.

Step-by-step explanation:

In a cosine function, the phase shift represents how much the function is shifting to the right or left. A positive phase shift indicates a shift to the right, while a negative phase shift indicates a shift to the left. The vertical translation refers to how the graph of the function is shifted up or down. The range of a function is the set of all possible y-values the function can take. For a cosine function, the range is typically -1 to 1.

For example, if we have the function f(x) = cos(x + π/2), the phase shift is π/2 to the left, and the graph is shifted up by π/2 units. The range of this function is still -1 to 1.

Answer 2

In sine and cosine functions, we have the following forms:

`\begin{gathered} f\mleft(x\mright)=A\sin\mleft(Bx+C\mright)+D \\ f\mleft(x\mright)=A\cos\mleft(Bx+C\mright)+D \end{gathered}`

Where A is the amplitute, 2π/B is the period, C is the phase shift and D is the vertical shift.

By comparison, we can see that:

`\begin{gathered} f\mleft(x\mright)=A\sin\mleft(Bx+C\mright)+D \\ f\mleft(x\mright)=\sin\mleft(x+45°\mright)+2 \end{gathered}`
`\begin{gathered} A=1 \\ B=1 \\ C=45° \\ D=2 \end{gathered}`

Then, the phase shift is 45°, the vertical shift is 2.

The vertical shift is the same as the middle horizontal axis of the function, so we know that the middle of the function is y = 2. The amplitute is how many units the function varies up and down from the middle. Since the Amplitute is 1, the function varies from 2 - 1 to 2 + 1, that is, it varies from 1 to 3. So, the range of the function is:

`R=\left[1,3\right]`