To determine the correct equation for the parabolic function described, let's analyze the options and the characteristics of a parabola that opens upwards.
The general form of a parabolic equation can be expressed as:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola and \(a\) determines the width and direction (upward if \(a > 0\) and downward if \(a < 0\)).
From the options given, we can examine the following:
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Vertex: From the structure of the equations, it appears that the vertex for each equation is located at \(x = 5\) since all options are of the form \((x - 5)^2\) or \((x - 3)^2\).
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Concavity: Since it states that the parabola is "concave up", we know that \(a\) must be positive (greater than 0).
From this, we can eliminate the option:
- \(y = - (x - 3)^2 - 5\) because it opens downwards (negative \(a\)).
Now, we focus on the options left:
- \(y=3(x−5)²−3\)
- \(y=2(x−5)²−3\)
- \(y=(x−5)²−3\)
Next, we need to consider the vertex and position of the parabola given that it passes through the 1st and 4th quadrants:
- The vertex appears to be at \((5, -3)\) since both key parameters match.
Now let's check if one of the equations agrees with this placement. We can find that, as \(x\) increases or decreases away from 5, \(y\) will be higher in the positive direction (increasing) due to its upward orientation.
Given that all options pass through the vertex at \((5, -3)\) but have different "a" values:
- If \(a = 1\), the parabola will be more narrow or broader depending on how far \(y\) moves from \(-3\) as \(x\) moves away from \(5\).
Considering the characteristics of the parabola, the simplest answer based on the common forms is typically assumed to be the more standard equation with \(a = 1\):
So, the equation you're looking for would most likely be:
Answer: \(y = (x - 5)^2 - 3\).
3 answers