To determine the correct equation for the concave up parabola based on the given options, we need to consider the general form of a parabola, which can be expressed 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.
From the information given, we know that the parabola is concave up, which tells us that \(a\) must be positive.
Analyzing the Options:
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Option 1: \(y = 3(x - 5)^2 - 3\)
- This is concave up (since \(a = 3 > 0\)) and has vertex at \((5, -3)\).
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Option 2: \(y = (x - 5)^2 - 3\)
- This is concave up (since \(a = 1 > 0\)) and has vertex at \((5, -3)\).
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Option 3: \(y = -(x - 3)^2 - 5\)
- This is concave down (since \(a = -1 < 0\)), so it cannot be correct.
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Option 4: \(y = 2(x - 5)^2 - 3\)
- This is concave up (since \(a = 2 > 0\)) and has vertex at \((5, -3)\).
Choosing the Equation:
Since the parabola is concave up and passes through the 1st and 4th quadrants, we need to determine which of the concave up options fits the description the best.
Based on the vertex location being at \((5, -3)\), it seems that both options 1, 2, and 4 could be viable candidates, but ultimately we also need to consider the scale and shape of the parabola when it intersects the axes.
- \(y = 3(x - 5)^2 - 3\) has a vertex and opens upwards but has a narrower shape.
- \(y = (x - 5)^2 - 3\) also has the vertex point but is wider than the previous.
- \(y = 2(x - 5)^2 - 3\) is between the two in terms of steepness.
Without seeing the graph, it is difficult to choose definitively. However, based solely on the requirement that it must pass through Quadrants 1 and 4 and the indications of the vertex being at \((5, -3)\):
The best choices for concavity and vertex location come from:
- Option 2: \(y = (x - 5)^2 - 3\) (wider parabolic opening).
- Option 4: \(y = 2(x - 5)^2 - 3\) (narrower).
Conclusion:
Based on standard characteristics and shapes of parabolas, the appropriate choice would be
y = (x - 5)^2 - 3.
However, if considering steepness, and it does seem to indicate a broader concavity, Option 4 could be equally valid without more context.
Thus, I'd summarize:
- If only concavity and vertex matter, Option 2 is a very likely candidate. If it leans towards a narrower form fitting the quadrants, Option 4 could work. But without the visual confirmation, Option 2 is the safest clear choice given the statements.