To determine the correct equation for the concave up parabola that you described, we can analyze the given options based on the structure of a parabola in vertex form, which is generally written as:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola and \(a\) determines the direction and width of the parabola.
Given that the parabola is concave up, \(a\) must be positive. We also assume the vertex lies between the quadrants given that it is passing through the 1st and 4th quadrants, influencing our choices based on the vertex and general shape.
Let's evaluate the options:
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\(y=−(x−3)²−5\): This is a downward-opening parabola because of the negative \(a\). Therefore, it cannot be correct.
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\(y=3(x−5)²−3\): This parabola opens upwards (since \(a = 3 > 0\)). The vertex is at \((5, -3)\). This could potentially fit, but we need to ensure that it intersects appropriate quadrants.
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\(y=(x−5)²−3\): This also opens upwards with the vertex at \((5, -3)\), and it shares the same vertex behavior as option 2.
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\(y=2(x−5)²−3\): This is similarly an upward-opening parabola (since \(a = 2 > 0\)) and also has its vertex at \((5, -3)\).
Now, we need to see where each one passes through based on the range of the x and y axes. The only conditions to check are if they intersect with the x-axis or go towards the quadrants specified.
Both option 2, 3, and 4 open upwards. The vertex position would suggest they stay within the bounds of the graph. The significant difference between them is their steepness (the value of \(a\)).
Since there is no specific information about the steepness or horizontal translation that immediately disqualifies any of them based purely on the graphical constraints described in your text:
- Both \(y=(x−5)²−3\) and \(y=2(x−5)²−3\) would generally fit the description as they go through the relevant quadrants given the shape and configuration in standard context.
In conclusion, without additional visual confirmation of intersections at specific points, for a potential single answer and considering simple upward motion, the most straightforward equation, generally reaching coordinates without loss, is:
\(y=(x−5)²−3\).
If the parabola must also be steeper, then that would mean choosing option 4 instead. Without clarity, option 3 is a sound choice given the context.
So the answer is:
\(y=(x−5)²−3\).