To determine the correct equation for the function represented by the parabola, let's analyze the possible equations presented along with their characteristics.
The function is concave up, which means it opens upwards. Therefore, any equation with a negative leading coefficient can be eliminated.
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y=−(x−3)²−5: This equation has a negative leading coefficient, so it opens downwards. Not a candidate.
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y=(x−5)²−3: This equation has a positive leading coefficient, so it opens upwards. It has its vertex at (5, -3).
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y=3(x−5)²−3: This equation also opens upwards (positive leading coefficient) and has its vertex at (5, -3) but is narrower than the previous one due to the factor of 3.
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y=2(x−5)²−3: This also opens upwards (positive leading coefficient) and has its vertex at (5, -3) but is wider than the equation with the factor of 3.
Given that the parabola passes through the 1st and 4th quadrants and opens upwards, both y=(x−5)²−3 and y=2(x−5)²−3 are possible candidates. The vertex is at (5, -3), and the equations suggest that this point is indeed below the x-axis and opens upwards toward the 1st quadrant.
If we don't have further details about the exact shape or dimensions of the parabola, any of these options could fit based on the information provided, but the simplest and likely correct choice given it is a standard upward opening parabola is:
y=(x−5)²−3.
This matches the characteristics of a concave up parabola passing through the specified quadrants.