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 negative k right parenthesis
(−4,−1)
left parenthesis negative h comma k right parenthesis
(4,−1)
left parenthesis h comma k right parenthesis
(−4,1)
?(1 point)
Responses
(4,1)
left parenthesis h comma negative k right parenthesis
(−4,−1)
left parenthesis negative h comma k right parenthesis
(4,−1)
left parenthesis h comma k right parenthesis
(−4,1)
Answers
GPT-4o mini
To find the ordered pair of the endpoint in the standard form of a square root function given by
\[
y = \sqrt{5x - 4} - 1,
\]
we start by rewriting this function to identify the vertex (endpoint) of the square root function.
### Standard Form Identification
The general form of a square root function can be represented as:
\[
y = a \sqrt{x - h} + k,
\]
where \((h, k)\) represents the vertex (or endpoint) of the function.
### Determine \(h\) and \(k\)
In our function, we have:
\[
y = \sqrt{5x - 4} - 1.
\]
To rewrite this in the standard form, we set the expression inside the square root equal to zero to find \(h\):
\[
5x - 4 = 0 \implies 5x = 4 \implies x = \frac{4}{5}.
\]
Substituting \(x = \frac{4}{5}\) into the function to find the corresponding \(y\)-value:
\[
y = \sqrt{5 \left(\frac{4}{5}\right) - 4} - 1 = \sqrt{4 - 4} - 1 = \sqrt{0} - 1 = -1.
\]
Thus, the vertex (endpoint) of the function is at:
\[
\left(\frac{4}{5}, -1\right).
\]
### Comparison with Given Options
Looking at the options provided:
1. (4, 1)
2. (−4,−1)
3. (4,−1)
4. (−4, 1)
None of the options match \(\left(\frac{4}{5}, -1\right)\).
However, we typically look for the transformation points or calculated values with respect to translations. The square root function is translated 1 unit down, meaning that the vertex moves down from point \(\left(\frac{4}{5}, 0\right)\) to \(\left(\frac{4}{5}, -1\right)\). The transformation does not yield any of the suggested options thus evaluating them rests outside standard form conventions. If ignoring the vertex as an option, further analysis would be required.
### Conclusion
Given the standard form and comparison:
The ordered pair of the endpoint from the function does not exactly correlate with offered options based on standard values. The closest response concerning signs and visual interpretations is notably:
**None of the options are the correct answer**, as we derived \( \left( \frac{4}{5}, -1 \right) \). If focusing purely on translation formats from transformations in geometry it may require a reevaluation of endpoint descriptions provided.
\[
y = \sqrt{5x - 4} - 1,
\]
we start by rewriting this function to identify the vertex (endpoint) of the square root function.
### Standard Form Identification
The general form of a square root function can be represented as:
\[
y = a \sqrt{x - h} + k,
\]
where \((h, k)\) represents the vertex (or endpoint) of the function.
### Determine \(h\) and \(k\)
In our function, we have:
\[
y = \sqrt{5x - 4} - 1.
\]
To rewrite this in the standard form, we set the expression inside the square root equal to zero to find \(h\):
\[
5x - 4 = 0 \implies 5x = 4 \implies x = \frac{4}{5}.
\]
Substituting \(x = \frac{4}{5}\) into the function to find the corresponding \(y\)-value:
\[
y = \sqrt{5 \left(\frac{4}{5}\right) - 4} - 1 = \sqrt{4 - 4} - 1 = \sqrt{0} - 1 = -1.
\]
Thus, the vertex (endpoint) of the function is at:
\[
\left(\frac{4}{5}, -1\right).
\]
### Comparison with Given Options
Looking at the options provided:
1. (4, 1)
2. (−4,−1)
3. (4,−1)
4. (−4, 1)
None of the options match \(\left(\frac{4}{5}, -1\right)\).
However, we typically look for the transformation points or calculated values with respect to translations. The square root function is translated 1 unit down, meaning that the vertex moves down from point \(\left(\frac{4}{5}, 0\right)\) to \(\left(\frac{4}{5}, -1\right)\). The transformation does not yield any of the suggested options thus evaluating them rests outside standard form conventions. If ignoring the vertex as an option, further analysis would be required.
### Conclusion
Given the standard form and comparison:
The ordered pair of the endpoint from the function does not exactly correlate with offered options based on standard values. The closest response concerning signs and visual interpretations is notably:
**None of the options are the correct answer**, as we derived \( \left( \frac{4}{5}, -1 \right) \). If focusing purely on translation formats from transformations in geometry it may require a reevaluation of endpoint descriptions provided.