To determine the correct equation for the concave up parabola that passes through the 1st and 4th quadrants, we need to examine the general form of the quadratic function, which is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and the width of the parabola.
Given that the parabola is concave up, \(a\) must be positive.
Let's analyze each response:
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\(y=(x−5)^2−3\): This has a vertex at (5, -3). Since -3 is not within the range of the y-axis (which goes from -4 to 2), this parabola does pass through the 4th quadrant but does not go up into the 1st quadrant, so this cannot be the correct equation.
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\(y=−(x−3)^2−5\): This function has \(a\) negative, indicating that it is concave down, which contradicts the requirement for the parabola to be concave up.
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\(y=3(x−5)^2−3\): This parabola has a vertex at (5, -3). Similar to option 1, it dips down below -4 and does not extend into the 1st quadrant, so this cannot be the correct equation.
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\(y=2(x−5)^2−3\): The vertex is again at (5, -3). Like the previous options, this does not satisfy the condition for passing through the 1st quadrant.
Since the 1st and 4th quadrants typically imply that the parabola must rise in both of these areas, and considering the vertex and the nature of the equations provided, the first choice is the only one with sufficient viability upon consideration. It specifies a concave up parabola even though its vertex location needs further inspection.
Ultimately, as the options fall short of meeting the criteria well, none fulfill the conditions for both 1st and 4th quadrants adequately without falling short on trying exact values on the plot.
However, the most reasonable conclusion based on the quick examination of the equations aligns with assuming the first response, albeit realizing that geometric interpretation likely limits optimal alignments with described axes.