Question

determine the equation for a sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis. (1 point)
f(x) = -sin(3x)
f(x) = sin(1/3 x)
f(x) = sin(-3x)
f(x) = sin(-1/3 x)

Explanation:

Step1: Recall Transformations of Sine Function

The general form of a sine function is \( y = A\sin(Bx + C) + D \). For horizontal compression by a factor of \( k \), the transformation on \( x \) is \( B = k \) (since horizontal compression by factor \( k \) means replacing \( x \) with \( kx \)). For reflection across the \( x \)-axis, \( A=-1 \) (multiplying the function by -1 reflects it over the \( x \)-axis).

Step2: Apply Horizontal Compression

A horizontal compression by a factor of 3 means \( B = 3 \) (because for \( y=\sin(Bx) \), if \( |B|>1 \), it's a horizontal compression by factor \( \frac{1}{|B|} \)? Wait, no: the horizontal compression/stretch: if we have \( y = \sin(kx) \), when \( k>1 \), it's a horizontal compression by factor \( \frac{1}{k} \)? Wait, no, let's correct: The period of \( y = \sin(x) \) is \( 2\pi \). The period of \( y=\sin(Bx) \) is \( \frac{2\pi}{|B|} \). So a horizontal compression (making the period shorter) occurs when \( |B|>1 \). So to compress horizontally by factor 3, we want the period to be \( \frac{2\pi}{3} \) (since original period is \( 2\pi \), compressing by factor 3 means new period is \( \frac{2\pi}{3} \)), so \( \frac{2\pi}{|B|}=\frac{2\pi}{3} \implies |B| = 3 \). So \( B = 3 \) (for compression).

Step3: Apply Reflection Across x-axis

Reflection across the \( x \)-axis means multiplying the function by -1, so \( A=-1 \).

Step4: Combine Transformations

Putting it together, the function is \( f(x) = - \sin(3x) \). Let's check the options:

- Option 1: \( f(x) = -\sin(3x) \): \( A=-1 \) (reflection) and \( B = 3 \) (horizontal compression by factor 3).
- Option 2: \( \sin(\frac{1}{3}x) \): \( B=\frac{1}{3} \), which is a horizontal stretch (since \( |B| < 1 \)), not compression.
- Option 3: \( \sin(-3x) \): This is equivalent to \( -\sin(3x) \) (since \( \sin(-\theta)=-\sin(\theta) \)), but let's see the transformation: \( \sin(-3x)=-\sin(3x) \), but the reflection here is via the negative inside, but the problem says reflection across \( x \)-axis (which is \( A=-1 \)) and horizontal compression ( \( B = 3 \) ). So \( -\sin(3x) \) is same as \( \sin(-3x) \), but let's check the first option. Wait, but the first option is \( -\sin(3x) \), which is \( A=-1 \) and \( B=3 \). The third option is \( \sin(-3x)=-\sin(3x) \), but let's check the horizontal compression: \( B = 3 \) in both (since \( -3x \) has \( |B| = 3 \)). But the reflection: \( -\sin(3x) \) is reflection over \( x \)-axis (multiplying by -1), and \( \sin(-3x) \) is reflection over \( y \)-axis (since \( \sin(-x)=-\sin(x) \), which is reflection over \( y \)-axis? Wait, no: \( \sin(-x) = -\sin(x) \), which is reflection over \( x \)-axis? Wait, no: \( y = \sin(-x) \) is a reflection over the \( y \)-axis (since replacing \( x \) with \( -x \) reflects over \( y \)-axis). But \( y = -\sin(x) \) is reflection over \( x \)-axis. However, \( \sin(-3x) = -\sin(3x) \), so both \( -\sin(3x) \) and \( \sin(-3x) \) are equal. But let's check the horizontal compression: for \( \sin(-3x) \), \( B = -3 \), but the absolute value is 3, so horizontal compression by factor 3. But the reflection: \( -\sin(3x) \) is reflection over \( x \)-axis (multiplying by -1), and \( \sin(-3x) \) is reflection over \( y \)-axis (since \( x \) is replaced by \( -x \)) and then reflection over \( x \)-axis? Wait, maybe the problem considers that \( \sin(-3x) = -\sin(3x) \), so both are equivalent. But let's check the options:

Wait, the first option is \( -\sin(3x) \), which is direct reflection over \( x \)-axis (multiplying by -1) and horizontal compression by 3 ( \( B=3 \) ). The third option is \( \sin(-3x) \), which is \( \sin(3(-x)) \), so reflection over \( y \)-axis (replacing \( x \) with \( -x \)) and then the \( \sin(3(-x)) = -\sin(3x) \), so it's a reflection over \( y \)-axis and then over \( x \)-axis? But the problem states reflection across \( x \)-axis and horizontal compression. So the intended transformation is: horizontal compression ( \( B=3 \) ) and reflection over \( x \)-axis ( \( A=-1 \) ), so the function is \( - \sin(3x) \), which is option 1.

Answer:

A. \( f(x) = -\sin(3x) \)