Question

Which function would represent a horizontal compression of j(x)=2x−4−−−−√+1 by a factor of 15 and reflection over the line x=4 ?(1 point) Responses m(x)=2−15(x−4)−−−−−−−−−√+1 m left parenthesis x right parenthesis equals 2 Start Root negative Start Fraction 1 over 5 End Fraction left parenthesis x minus 4 right parenthesis End Root plus 1 w(x)=−10x−4−−−−−√+1 w left parenthesis x right parenthesis equals negative 10 Start Root x minus 4 End Root plus 1 k(x)=2−5(x−4)−−−−−−−−−√+1 k left parenthesis x right parenthesis equals 2 Start Root negative 5 left parenthesis x minus 4 right parenthesis End Root plus 1 q(x)=−2x−9−−−−−√+1

Answer 1

To create a function that represents a horizontal compression and reflection of the provided function \( j(x) = \sqrt{2x - 4} + 1 \), we need to follow these transformations step by step.

1. Horizontal Compression by a factor of 15: A horizontal compression by a factor of \( k \) involves substituting \( x \) by \( \frac{x}{k} \). For a factor of 15, this means we replace \( x \) with \( 15x \).

2. Reflection over the line \( x = 4 \): To reflect a function over the line \( x = 4 \), we need to translate the function to the line of reflection, reflect it, and then translate it back. The reflection can be achieved by modifying our variable as follows:

   • Translate \( x \) to \( x-4 \),
   • Reflect it by negating the expression, and
   • Translate back.

Translating \( x \) to be centered around 4:

• Reflected point would be \( 4 - (x - 4) = 8 - x \).
• After inserting it into the function, the transformations combine to yield the new function.

Combining these transformations, we derive:

• Starting with \( j(x) = \sqrt{2x - 4} + 1 \),
• Applying horizontal compression: \( j(15x) = \sqrt{2(15x) - 4} + 1 = \sqrt{30x - 4} + 1 \),
• Reflecting this point around \( x = 4 \): Replace \( x \) with \( 8 - x \) in the horizontally compressed equation.

Now, substituting \( x \) in the function:

\[ j(15(8 - x)) = \sqrt{30(8 - x) - 4} + 1 \]

This yields the transformation.

After going over the answer options, you would look for the one that matches the required transformation based on the adjustments we performed.

Given the selections, let's analyze each one:

• m(x) includes -15, which appears to go against our reflection.

• k(x) appears as a promising candidate with the term involving \( -5 \) in the square root.

• w(x) contains the right transformations with negations and reflects appropriately.

After thorough analysis, the function that denotes both the horizontal compression of the original AND the reflection over \( x = 4 \), based on the provided options will be:

\[ m(x) = 2\sqrt{-\frac{1}{5}(x-4)} + 1 \]

While not ideal, the key to solving would lie in recognizing structural adjustments. A synthesis check against existing forms did suggest \( w \) being the response should undergo a reevaluation after reflective adjustments.

Conclusively, ensure you can align simple transformations of compositions in navigating these equations if matching exists directly.