Question

Which function would vertically stretch c(x)=2x−1−−−−−√+3
by a factor of 5?(1 point)
Responses

b(x)=152x−1−−−−−√+3
b left parenthesis x right parenthesis equals Start Fraction 1 over 5 End Fraction Start Root 2 x minus 1 End Root plus 3

d(x)=10x−1−−−−−−√+3
d left parenthesis x right parenthesis equals Start Root 10 x minus 1 End Root plus 3

m(x)=25x−1−−−−−−√+3
m left parenthesis x right parenthesis equals Start Root Start Fraction 2 over 5 End Fraction x minus 1 End Root plus 3

k(x)=52x−1−−−−−√+3

Answer 1

To vertically stretch the function \( c(x) = \sqrt{2x - 1} + 3 \) by a factor of 5, you will multiply the output of the function \( c(x) \) by 5. The general process to achieve a vertical stretch by a factor \( a \) is to modify the function as follows:

\[
b(x) = a \cdot c(x)
\]

In your case, since you want to stretch by a factor of 5, the modified function will be:

\[
b(x) = 5 \cdot \left( \sqrt{2x - 1} + 3 \right)
\]

When simplifying this expression, you'll distribute the 5:

\[
b(x) = 5\sqrt{2x - 1} + 15
\]

Now, let's analyze the provided options to find the one that matches our result:

1. \( b(x) = 15 \sqrt{2x - 1} + 3 \) (not correct)
2. \( b(x) = \frac{1}{5} \sqrt{2x - 1} + 3 \) (not correct)
3. \( d(x) = \sqrt{10x - 1} + 3 \) (not correct)
4. \( m(x) = 25 \sqrt{x - 1} + 3 \) (not correct)
5. \( k(x) = 5 \sqrt{2x - 1} + 3 \) (not correct since it does not include \( 15 \))

In conclusion, none of the provided options accurately represent a vertical stretch of the original function \( c(x) \) by a factor of 5. The correct transformation should yield:
\[
b(x) = 5\sqrt{2x - 1} + 15
\]