To determine the correct equation of the parabolic function given the description, we need to analyze the options based on the standard form of a parabola, which is \( y = a(x-h)^2 + k \), where (h, k) is the vertex of the parabola and 'a' determines the width and direction (concave up or down).
If the parabola is concave up and passes through the 1st and 4th quadrants, it will open upwards.
Let's evaluate the provided options for their constituents:
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Based on the concavity: The first three options have positive leading coefficients (concave up), while the second option, \( y = - (x-3)^2 - 5 \), would open downward.
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Finding the vertex: The vertex of the given options can be extracted from their forms:
- For \( y = (x-5)^2 - 3 \), the vertex is at \( (5, -3) \).
- For \( y = 2(x-5)^2 - 3 \), the vertex is still at \( (5, -3) \) but stretches it vertically.
- For \( y = 3(x-5)^2 - 3 \), it also has a vertex at \( (5, -3) \).
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Analyzing the range: The parabola is said to extend into the 1st and 4th quadrants, indicating it must drop below and rise above the x-axis. Since the y-axis goes from -4 to 2 based on your description, we need to ensure that the function can move through those quadrants within the specified range.
As the option \( y = (x-5)^2 - 3 \) opens upwards, we notice it has a vertex below the x-axis. This allows it to rise into the first quadrant when \( x < 5 \) and fall into the fourth quadrant as it goes past \( x = 5 \).
To find if specific y-values for x=0 or x=8 fit within the 1st and 4th quadrants, calculate the outputs for each proposed function.
Testing:
- \( y = (0-5)^2 - 3 = 25 - 3 = 22 \) (above x-axis)
- \( y = (8-5)^2 - 3 = 9 - 3 = 6 \) (above x-axis)
Thus, the outputs for other options either lead to similarly fitting scenarios or yields outputs too far below y = -4 for their maximum possible x ranges.
Final conclusion: The function that fits all criteria presented in the original prompt while being concave up and passing through relevant x and y ranges, specifically the 1st and 4th quadrants, is:
\( y = (x-5)^2 - 3 \).
3 answers