Question
Which of the following correctly identifies the quadratic function that is related to the square root function k(x)=5x−−√?(1 point)
Responses
m(x)=5x2
m left parenthesis x right parenthesis equals 5 x squared
m(x)=125x2
m left parenthesis x right parenthesis equals Start Fraction 1 over 25 End Fraction x squared
m(x)=25x2
m left parenthesis x right parenthesis equals 25 x squared
m(x)=15x2
Responses
m(x)=5x2
m left parenthesis x right parenthesis equals 5 x squared
m(x)=125x2
m left parenthesis x right parenthesis equals Start Fraction 1 over 25 End Fraction x squared
m(x)=25x2
m left parenthesis x right parenthesis equals 25 x squared
m(x)=15x2
Answers
GPT-4o mini
To find the quadratic function related to the square root function \( k(x) = \sqrt{5x} \), we can square both sides of the equation. If we let \( m(x) \) be the quadratic function, we have:
1. Start with the square root function:
\[ k(x) = \sqrt{5x} \]
2. Square both sides to eliminate the square root:
\[ k(x)^2 = 5x \]
3. This implies:
\[ m(x) = (k(x))^2 = 5x \]
Rearranging this for a standard quadratic form, we note that \( 5x \) relates to \( m(x) \).
We want \( m(x) \) to be expressed in terms of \( x^2 \). The quadratic function related to \( k(x) \) in terms of \( x \) must actually involve \( c = 5 \) as a scaling factor.
Looking at the options provided:
- \( m(x) = 5x^2 \) --> This would lead back to the original function after taking the square root.
- \( m(x) = 125x^2 \) --> Not related directly.
- \( m(x) = \frac{1}{25} x^2 \) --> Not relevant since it doesn't match forms.
- \( m(x) = 25x^2 \) --> This would also lead back incorrectly to roots.
- \( m(x) = 15x^2 \) --> Again does not correspond.
The correct relationship is between the original square root transformation to get to \( m(x) = 5x^2 \) given that if we wanted a simplified formulation closer to root.
### Conclusion:
The quadratic function related to \( k(x) = \sqrt{5x} \) is closest to:
**\( m(x) = 5x^2 \)**.
1. Start with the square root function:
\[ k(x) = \sqrt{5x} \]
2. Square both sides to eliminate the square root:
\[ k(x)^2 = 5x \]
3. This implies:
\[ m(x) = (k(x))^2 = 5x \]
Rearranging this for a standard quadratic form, we note that \( 5x \) relates to \( m(x) \).
We want \( m(x) \) to be expressed in terms of \( x^2 \). The quadratic function related to \( k(x) \) in terms of \( x \) must actually involve \( c = 5 \) as a scaling factor.
Looking at the options provided:
- \( m(x) = 5x^2 \) --> This would lead back to the original function after taking the square root.
- \( m(x) = 125x^2 \) --> Not related directly.
- \( m(x) = \frac{1}{25} x^2 \) --> Not relevant since it doesn't match forms.
- \( m(x) = 25x^2 \) --> This would also lead back incorrectly to roots.
- \( m(x) = 15x^2 \) --> Again does not correspond.
The correct relationship is between the original square root transformation to get to \( m(x) = 5x^2 \) given that if we wanted a simplified formulation closer to root.
### Conclusion:
The quadratic function related to \( k(x) = \sqrt{5x} \) is closest to:
**\( m(x) = 5x^2 \)**.