Question

Question
The graph of p(x)=√x+5−9
needs to be horizontally compressed by a factor of 15
. What should the value of k be in the new function g(x)=√k(x+5)−9
?(1 point)
The value of k should be

Answer 1

To horizontally compress the function \( p(x) = \sqrt{x + 5} - 9 \) by a factor of 15, we need to replace \( x \) in the function with \( \frac{x}{k} \) where \( k \) is the factor by which we multiply \( x \) inside the square root.

For a horizontal compression by a factor of \( c \), we substitute \( x \) with \( \frac{x}{c} \). In this case, \( c = 15 \), which means we should substitute \( x \) in the original function \( p(x) \) as follows:

\[
p\left(\frac{x}{15}\right) = \sqrt{\frac{x}{15} + 5} - 9
\]

To express the new function \( g(x) \) in the form \( g(x) = \sqrt{k(x + 5)} - 9 \), we want to rewrite the inside of the square root.

1. Start from the expression:

\[
\sqrt{\frac{x}{15} + 5} = \sqrt{\frac{x + 75}{15}}
\]

This implies:

\[
\sqrt{\frac{1}{15}(x + 75)} = \frac{1}{\sqrt{15}} \sqrt{x + 75}
\]

2. To get our expression to match the form \( g(x) = \sqrt{k(x + 5)} - 9 \), we need to manipulate \( \frac{x + 75}{15} = \frac{1}{15}(x + 75) \) to fit into \( \sqrt{k(x + 5)} \).

Notice that \( x + 75 \) can be factored from \( 15 \) as \( 15 \cdot \frac{1}{15}(x + 75) \).

3. Now, we equate:

\[
k(x+5) = \frac{1}{15}(x + 75)
\]

To figure out \( k \), we need the models of the equations to match, aligning the coefficients of the \( x \) terms.

The term \( (x + 75) \) when we substitute \( x + 5 \), will require:

\[
k(x + 5) = kx + 5k \quad \will match \quad \frac{1}{15}x + \frac{75}{15}=\frac{1}{15}x + 5
\]

From comparing the two:

A condition for equality will be:

1. Coefficient of \( x \): \( k = \frac{1}{15} \)
2. Constant terms: \( 5k = 5 \Rightarrow k = 1 \)

So, simplest alignment leads us to:

Since both constant conditions imply that \( k = 1 \) provides the needed condition.

The horizontal compression factor corresponds that:
Thus,

The value of \( k \) should be:

\[
15
\]
So the function \( g(x) \) becomes \( \sqrt{15(x + 5)} - 9 \) ensuring correct compressive behavior around specified conditions.

Final response:

**The value of \( k \) should be 15.**