Analyze the two functions w(x)=23x−1−−−−−√+3

Question

𝑤
(
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)
=
2
3
𝑥
−
1
+
3
and m(x)=63x−1−−−−−√+3
𝑚
(
𝑥
)
=
6
3
𝑥
−
1
+
3
. What is the transformation that occurs from w(x)
𝑤
(
𝑥
)
to m(x)
𝑚
(
𝑥
)
?(1 point)
The transformation that occurs is a vertical stretch by a factor of

Answer 1

To analyze the functions \( w(x) = \sqrt{23x - 1} + 3 \) and \( m(x) = \sqrt{63x - 1} + 3 \), we can determine the transformation from \( w(x) \) to \( m(x) \).

Both functions share the same vertical shift of +3. The important part to analyze is the transformation within the square root.

Step 1: Transform the Expressions Inside the Square Root

1. For \( w(x) \): \[ w(x) = \sqrt{23x - 1} + 3 \]

2. For \( m(x) \): \[ m(x) = \sqrt{63x - 1} + 3 \]

Step 2: Compare the Two Functions

The critical part of the transformation occurs in the expressions under the square root:

• In \( w(x) \), we have \( 23x - 1 \).
• In \( m(x) \), we have \( 63x - 1 \).

Step 3: Analyze the Coefficient of \( x \)

To understand how the functions transform from one to another, we can analyze the coefficient of \( x \):

• The coefficient in \( w(x) \) is 23.
• The coefficient in \( m(x) \) is 63.

Step 4: Calculate the Vertical Stretch Factor

When moving from \( w(x) \) to \( m(x) \): \[ m(x) = w\left(\frac{63}{23}x\right) \] This indicates that \( m(x) \) is a result of a horizontal transformation of \( w(x) \) where we scale \( x \) by a factor of \( \frac{63}{23} \).

To find the vertical stretch factor, note that the function \( \sqrt{kx} \) exhibits a vertical stretch of \( k \) when \( k > 1 \).

Calculate the stretch factor: \[ \text{Vertical stretch factor} = \frac{63}{23} \]

Conclusion

The transformation from \( w(x) \) to \( m(x) \) is a vertical stretch by a factor of \( \frac{63}{23} \) (approximately 2.739).

So, the vertical stretch factor is \( \frac{63}{23} \).