Question

which function represents a vertical stretch by a factor of 1.85 with a reflection over the line ( y = -1 ) on function ( w(x) = -sqrt{3(x + 2)} - 1 )? (1 point)

( circ ) ( h(x) = \frac{sqrt{3(x + 2)}}{1.85} - 1 )

( circ ) ( t(x) = 1.85 + sqrt{3(x + 2)} - 1 )

( circ ) ( k(x) = sqrt{4.85(x + 2)} - 1 )

( circ ) ( r(x) = 1.85sqrt{3(x + 2)} - 1 )

Explanation:

Step1: Recall Vertical Stretch and Reflection Rules

For a function \( y = f(x) \), a vertical stretch by a factor of \( a \) is \( y = a \cdot f(x) \), and a reflection over the line \( y = k \) can be thought of as transforming the vertical distance from the line. First, let's analyze the original function \( w(x)=-\sqrt{3(x + 2)}-1 \).

Step2: Analyze Vertical Stretch

A vertical stretch by factor \( 1.85 \) on a function \( y = f(x) \) is \( y=1.85\cdot f(x) \) (if we consider the stretch first, then reflection). Wait, but we also have a reflection over \( y = - 1 \). Wait, maybe first, let's re - express the reflection over \( y=-1 \). The reflection of a point \( (x,y) \) over the line \( y = k \) is given by \( (x,2k - y) \). So for \( y = w(x) \), the reflection over \( y=-1 \) is \( y'=2(-1)-w(x)=-2 - w(x) \).

First, let's find \( w(x)=-\sqrt{3(x + 2)}-1 \). So \( -w(x)=\sqrt{3(x + 2)}+1 \), and \( -2 - w(x)=\sqrt{3(x + 2)}+1-2=\sqrt{3(x + 2)}-1 \)? Wait, no, maybe I made a mistake. Wait, the original function has a negative sign in front of the square root. Let's start over.

The original function is \( w(x)=-\sqrt{3(x + 2)}-1 \). Let's first handle the reflection over \( y = - 1 \). The formula for reflecting a function \( y = f(x) \) over the line \( y = k \) is \( y=2k - f(x) \). Here \( k=-1 \), so the reflected function is \( y = 2(-1)-w(x)=-2-(-\sqrt{3(x + 2)}-1)=-2+\sqrt{3(x + 2)} + 1=\sqrt{3(x + 2)}-1 \).

Now, we need to apply a vertical stretch by a factor of \( 1.85 \) to this reflected function. A vertical stretch of a function \( y = g(x) \) by a factor of \( a \) is \( y=a\cdot g(x) \). So if \( g(x)=\sqrt{3(x + 2)}-1 \)? Wait, no, wait the reflection step: Wait, maybe the order is stretch then reflect or reflect then stretch? Wait, the problem says "a vertical stretch by a factor of 1.85 with a reflection over the line \( y=-1 \) on function \( w(x) \)".

Wait, let's re - express the transformation steps. Let's first consider the vertical stretch. A vertical stretch of \( w(x) \) by factor \( 1.85 \) is \( 1.85\times w(x) \). Then, reflect over \( y = - 1 \). Wait, no, the problem says "a vertical stretch by a factor of 1.85 with a reflection over the line \( y=-1 \)". Let's parse the original function \( w(x)=-\sqrt{3(x + 2)}-1 \).

First, let's consider the reflection over \( y=-1 \). Let's take a point \( (x,y) \) on \( w(x) \), its reflection over \( y = - 1 \) is \( (x, - 2 - y) \). So \( y'=-2 - y=-2-(-\sqrt{3(x + 2)}-1)=-2+\sqrt{3(x + 2)} + 1=\sqrt{3(x + 2)}-1 \). Now, apply a vertical stretch by factor \( 1.85 \) to this reflected function. A vertical stretch of a function \( y = g(x) \) by factor \( a \) is \( y = a\times g(x) \). So \( g(x)=\sqrt{3(x + 2)}-1 \), then the stretched function is \( y = 1.85\times(\sqrt{3(x + 2)}-1) \)? No, that's not right. Wait, maybe the reflection is over the line \( y=-1 \), so we can think of shifting the function up by 1 unit, reflecting over the x - axis, then shifting down by 1 unit. Wait, shifting \( w(x) \) up by 1 unit: \( w(x)+1=-\sqrt{3(x + 2)} \). Reflecting over the x - axis (which is \( y = 0 \)): \( - (w(x)+1)=\sqrt{3(x + 2)} \). Then shifting down by 1 unit: \( \sqrt{3(x + 2)}-1 \). Now, apply a vertical stretch by factor \( 1.85 \) to this function. A vertical stretch of \( y = \sqrt{3(x + 2)}-1 \) by factor \( 1.85 \) is \( y=1.85\times\sqrt{3(x + 2)}-1.85\times1 \)? No, that's not matching the options. Wait, maybe the order is stretch first then reflect. Let's check the options.

Wait, the original function is \( w(x)=-\sqrt{3(x + 2)}-1 \). Let's analyze each option:

- Option 1: \( h(x)=\frac{\sqrt{3(x + 2)}}{1.85}-1 \): This is a vertical compression, not a stretch. So eliminate.
- Option 2: \( t(x)=1.85+\sqrt{3(x + 2)}-1= \sqrt{3(x + 2)}+0.85 \). Let's see, if we reflect \( w(x) \) over \( y=-1 \), we get \( \sqrt{3(x + 2)}-1 \) (from earlier), then a vertical stretch by 1.85? No, but let's check the third option.
- Option 3: \( k(x)=\sqrt{4.85(x + 2)}-1 \): This is a horizontal or inside the square - root stretch, not vertical. So eliminate.
- Option 4: \( r(x)=1.85\sqrt{3(x + 2)}-1 \): Wait, maybe I made a mistake in the reflection. Wait, the original function has a negative sign in front of the square root. Let's consider the reflection over \( y=-1 \) as changing the sign of the function relative to \( y = - 1 \). Let's rewrite \( w(x)=-\sqrt{3(x + 2)}-1 \), so \( w(x)+1=-\sqrt{3(x + 2)} \). Reflecting over \( y = 0 \) (the x - axis) gives \( \sqrt{3(x + 2)} \), then \( \sqrt{3(x + 2)}-1 \) (shifting down by 1). Now, a vertical stretch by factor \( 1.85 \) would be \( 1.85\times\sqrt{3(x + 2)}-1 \), which is \( r(x) \). Wait, but let's check the reflection again. Alternatively, maybe the problem means reflection over the x - axis (but it says \( y=-1 \)). Wait, maybe the question has a typo, or my reflection formula is wrong. But among the options, the vertical stretch by factor 1.85 of the reflected (over \( y=-1 \)) function. The original function's negative sign in front of the square root: reflecting over \( y=-1 \) would make the square root positive (since \( - (-\sqrt{...})=\sqrt{...} \) relative to \( y=-1 \)), then stretching vertically by 1.85. So the function after reflection (making the square root positive) is \( \sqrt{3(x + 2)}-1 \), then vertical stretch by 1.85 is \( 1.85\sqrt{3(x + 2)}-1 \), which is option \( r(x) \). Wait, but let's check the options again. Wait, option \( t(x)=1.85+\sqrt{3(x + 2)}-1=\sqrt{3(x + 2)} + 0.85 \), option \( r(x)=1.85\sqrt{3(x + 2)}-1 \). Wait, maybe the reflection over \( y=-1 \) is equivalent to reflecting over the x - axis and then shifting, but the key is the vertical stretch. A vertical stretch of a function \( y = f(x) \) by factor \( a \) is \( y=a\times f(x) \). If we first reflect the function \( w(x) \) over \( y=-1 \) to get a function with a positive square root (since \( w(x)=-\sqrt{...}-1 \), reflecting over \( y=-1 \) would flip the sign of the square - root part), so the reflected function (before stretch) is \( \sqrt{3(x + 2)}-1 \), then stretching vertically by 1.85 gives \( 1.85\sqrt{3(x + 2)}-1 \), which is \( r(x) \). Wait, but let's check the options again. The original function is \( w(x)=-\sqrt{3(x + 2)}-1 \). Let's apply the transformations:

1. Vertical stretch by factor 1.85: Multiply the function by 1.85: \( 1.85\times w(x)=1.85\times(-\sqrt{3(x + 2)}-1)=-1.85\sqrt{3(x + 2)}-1.85 \)
2. Reflect over \( y=-1 \): The formula for reflecting \( y = f(x) \) over \( y = k \) is \( y = 2k - f(x) \). So \( y=2(-1)-(-1.85\sqrt{3(x + 2)}-1.85)=-2 + 1.85\sqrt{3(x + 2)}+1.85=1.85\sqrt{3(x + 2)}-0.15 \). No, that's not matching.

Wait, maybe the problem means reflection over the x - axis ( \( y = 0 \)) instead of \( y=-1 \). If we reflect \( w(x) \) over the x - axis, we get \( -w(x)=\sqrt{3(x + 2)}+1 \), then vertical stretch by 1.85: \( 1.85\times(\sqrt{3(x + 2)}+1)=1.85\sqrt{3(x + 2)}+1.85 \), which is not an option.

Wait, let's look at the options again. The option \( r(x)=1.85\sqrt{3(x + 2)}-1 \). The original function has \( -\sqrt{3(x + 2)}-1 \). If we remove the negative sign (which could be a reflection over \( y=-1 \) or over the x - axis), and then apply a vertical stretch by 1.85, we get \( 1.85\sqrt{3(x + 2)}-1 \). Among the given options, this is the only one that represents a vertical stretch (multiplying the square - root part by 1.85) after a reflection (removing the negative sign, which could be the reflection over \( y=-1 \) or x - axis). The other options: \( h(x) \) is a division by 1.85 (compression), \( t(x) \) is adding 1.85, \( k(x) \) is changing the inside of the square root. So the most probable answer is \( r(x) \).

Answer:

\( \boldsymbol{r(x) = 1.85\sqrt{3(x + 2)}-1} \) (the option corresponding to \( r(x) = 1.85\sqrt{3(x + 2)}-1 \))