Question
What is the ordered pair of the endpoint in the standard form of a square root function y=5x−4−−−−√−1
?(1 point)
Responses
(4,−1)
left parenthesis h comma k right parenthesis
(−4,−1)
left parenthesis negative h comma k right parenthesis
(−4,1)
left parenthesis negative h comma negative k right parenthesis
(4,1)
?(1 point)
Responses
(4,−1)
left parenthesis h comma k right parenthesis
(−4,−1)
left parenthesis negative h comma k right parenthesis
(−4,1)
left parenthesis negative h comma negative k right parenthesis
(4,1)
Answers
GPT-4o mini
To determine the ordered pair of the endpoint in the standard form of the square root function \( y = \sqrt{5x - 4} - 1 \), let's rewrite this in a way that highlights its vertex or endpoint.
The standard form of a square root function is generally expressed as:
\[
y = a \sqrt{x - h} + k
\]
where \( (h, k) \) is the vertex of the function.
### Analyzing the Given Function
From \( y = \sqrt{5x - 4} - 1 \), we can rewrite it as:
\[
y = \sqrt{5(x - \frac{4}{5})} - 1
\]
Here we want to identify \( h \) and \( k \). To do this, we can see that:
- \( 5x - 4 = 5(x - \frac{4}{5}) \), indicating that \( h = \frac{4}{5} \).
- \( k = -1 \) because of the \(-1\) at the end.
Thus, in the endpoint format, \( (h, k) \) translates to:
\[
\left( \frac{4}{5}, -1 \right)
\]
### Finding the Endpoint
To find the endpoints, we need to know the values that \( x \) can take. The square root is defined when \( 5x - 4 \geq 0 \), leading to:
\[
5x \geq 4 \quad \Rightarrow \quad x \geq \frac{4}{5}
\]
So we can identify that the function opens right from this minimum \( x \).
### Corresponding Points
When \( x = \frac{4}{5} \):
\[
y = \sqrt{5\left(\frac{4}{5}\right) - 4} - 1 = \sqrt{0} - 1 = -1
\]
Indicating that one of the endpoints is indeed at:
\[
\left( \frac{4}{5}, -1 \right)
\]
### Checking the Responses
None of the available responses are exactly \(\left( \frac{4}{5}, -1 \right)\) but we can check backwards to see which pair corresponds to the transformation based on \( (h, k) \):
- \( (4, -1) \) does not match since \( h = \frac{4}{5} \).
- \( (-4, -1) \) does not match.
- \( (-4, 1) \) does not match.
- \( (4, 1) \) does not match.
### Conclusion
Given these transformations, the correct form for endpoints comes out of:
\[
h = \frac{4}{5} \text{ doesn't match exact pairs, yet \( k = -1 \) hints the structure.}
\]
If the ultimate result that matches the \( (h, k) \) transformation is an expected format, and combined `h` properly will lead to transformation addressing the four endpoints provided.
So, if you are left with \( (4, -1) \) based on possibility and setups for result options, that could be considered closest as amongst available choices since it's in connection with vertex placed accordingly.
### Answer:
The answer based in part is closest to \( (4, -1) \) when interpreting transformation consistencies.
The standard form of a square root function is generally expressed as:
\[
y = a \sqrt{x - h} + k
\]
where \( (h, k) \) is the vertex of the function.
### Analyzing the Given Function
From \( y = \sqrt{5x - 4} - 1 \), we can rewrite it as:
\[
y = \sqrt{5(x - \frac{4}{5})} - 1
\]
Here we want to identify \( h \) and \( k \). To do this, we can see that:
- \( 5x - 4 = 5(x - \frac{4}{5}) \), indicating that \( h = \frac{4}{5} \).
- \( k = -1 \) because of the \(-1\) at the end.
Thus, in the endpoint format, \( (h, k) \) translates to:
\[
\left( \frac{4}{5}, -1 \right)
\]
### Finding the Endpoint
To find the endpoints, we need to know the values that \( x \) can take. The square root is defined when \( 5x - 4 \geq 0 \), leading to:
\[
5x \geq 4 \quad \Rightarrow \quad x \geq \frac{4}{5}
\]
So we can identify that the function opens right from this minimum \( x \).
### Corresponding Points
When \( x = \frac{4}{5} \):
\[
y = \sqrt{5\left(\frac{4}{5}\right) - 4} - 1 = \sqrt{0} - 1 = -1
\]
Indicating that one of the endpoints is indeed at:
\[
\left( \frac{4}{5}, -1 \right)
\]
### Checking the Responses
None of the available responses are exactly \(\left( \frac{4}{5}, -1 \right)\) but we can check backwards to see which pair corresponds to the transformation based on \( (h, k) \):
- \( (4, -1) \) does not match since \( h = \frac{4}{5} \).
- \( (-4, -1) \) does not match.
- \( (-4, 1) \) does not match.
- \( (4, 1) \) does not match.
### Conclusion
Given these transformations, the correct form for endpoints comes out of:
\[
h = \frac{4}{5} \text{ doesn't match exact pairs, yet \( k = -1 \) hints the structure.}
\]
If the ultimate result that matches the \( (h, k) \) transformation is an expected format, and combined `h` properly will lead to transformation addressing the four endpoints provided.
So, if you are left with \( (4, -1) \) based on possibility and setups for result options, that could be considered closest as amongst available choices since it's in connection with vertex placed accordingly.
### Answer:
The answer based in part is closest to \( (4, -1) \) when interpreting transformation consistencies.