Question

which statement accurately describes how adding a number, n, to the function $f(x) = \sin(x)$ affects its graph?

• there is a vertical shift of n units.
• the x-intercepts will shift n units.
• there is a change in amplitude of n units.
• the range will change by a factor of 2n units.

Explanation:

Step1: Recall Transformations of Sine Function

The general form of a sine function transformation is \( f(x) = A\sin(Bx - C) + D \), where \( D \) represents vertical shift. For the function \( f(x)=\sin(x)+n \), this is a vertical shift (since \( n \) is added outside the sine function, similar to \( D \) in the general form).

Step2: Analyze Each Option

• Option 1 (Vertical shift of \( n \) units): When we add a constant \( n \) to \( \sin(x) \), the graph shifts vertically by \( n \) units (up if \( n>0 \), down if \( n<0 \)). This matches the vertical shift concept.
• Option 2 (x - intercepts shift \( n \) units): x - intercepts are related to horizontal shifts or solving \( \sin(x)+n = 0 \). A vertical shift does not directly shift x - intercepts by \( n \) units in a linear way. For example, if \( n = 1 \), \( \sin(x)+1 = 0\) has no solutions, so this is incorrect.
• Option 3 (Change in amplitude of \( n \) units): Amplitude is related to the coefficient of \( \sin(x) \) (the \( A \) in \( A\sin(x) \)). Adding \( n \) does not change the amplitude (amplitude of \( \sin(x) \) is 1, and adding \( n \) doesn't affect the coefficient of \( \sin(x) \)), so this is incorrect.
• Option 4 (Range changes by factor of \( 2n \) units): The range of \( \sin(x) \) is \([- 1,1]\). The range of \( \sin(x)+n \) is \([n - 1,n + 1]\), so the change in range is \( (n + 1)-(n - 1)=2 \), not a factor of \( 2n \). So this is incorrect.

Answer:

There is a vertical shift of \( n \) units.