To determine the correct equation of the concave up parabola that passes through the given characteristics, we need to analyze the possible options.
- For a parabola that opens upwards, the coefficient in front of the squared term must be positive.
- The vertex form of a parabola is typically expressed as \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex.
Based on the given equations:
- \(y=3(x−5)^{2}−3\) - This is concave up (positive coefficient).
- \(y=(x−5)^{2}−3\) - This is also concave up (positive coefficient).
- \(y=2(x−5)^{2}−3\) - This is also concave up (positive coefficient).
- \(y=−(x−3)^{2}−5\) - This is concave down (negative coefficient).
Since we are looking for a concave up parabola, the last option can be eliminated immediately.
Now we need to determine which of the first three is correct. Without the image for specific details such as vertex location and direction, it's hard to pinpoint the exact answer, but we can analyze the vertex based on typical options.
The equations with the vertex \((5, -3)\) are:
- \(y=3(x−5)^{2}−3\)
- \(y=(x−5)^{2}−3\)
- \(y=2(x−5)^{2}−3\)
The key features to observe would be the vertex, the width of the parabola (determined by \(a\)), and whether the point lies in the first or fourth quadrant as mentioned.
Since it's generally more typical to select the vertex positioning and functional behavior around \(x=5\) and \(y=-3\), without visual confirmation, I'd recommend choosing:
y=(x−5)²−3
This is the simplest form that suggests a standard parabola opening upwards and fits the conditions provided.
Choose y=(x−5)²−3.