To determine which set of parametric equations represents the function \( y = 4x^2 - 43x + 97 \), we need to express \( x \) in terms of a parameter \( t \) and then try to express \( y \) in terms of that same parameter such that it matches the original function.
Let’s analyze each option:
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Option 1: \[ x = t - 5 \ y = 4t^2 - 3t + 12 \] Substituting \( x = t - 5 \) gives \( t = x + 5 \). Therefore, \[ y = 4(x + 5)^2 - 3(x + 5) + 12 \] Expanding \( y \): \[ = 4(x^2 + 10x + 25) - 3x - 15 + 12 \ = 4x^2 + 40x + 100 - 3x - 15 + 12 \ = 4x^2 + 37x + 97 , (\text{not } -43x) \]
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Option 2: \[ x = 2t \ y = t^2 - 21t + 97 \] Thus, \( t = \frac{x}{2} \). Now substituting that in for \( y \): \[ y = \left(\frac{x}{2}\right)^2 - 21\left(\frac{x}{2}\right) + 97 \ = \frac{x^2}{4} - \frac{21x}{2} + 97 \] Multiplying everything by 4 to eliminate the fraction gives: \[ 4y = x^2 - 42x + 388 \text{ (not } 4x^2) \]
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Option 3: \[ x = t + 5 \ y = 4t^2 - 4t - 18 \] So, \( t = x - 5 \). Now substitute into \( y \): \[ y = 4(x - 5)^2 - 4(x - 5) - 18 \ = 4(x^2 - 10x + 25) - 4x + 20 - 18 \ = 4x^2 - 40x + 100 - 4x + 20 - 18 \ = 4x^2 - 44x + 102 , (\text{slightly off, not } -43x) \]
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Option 4: \[ x = t + 3 \ y = 4t^2 - 19t + 5 \] Thus, \( t = x - 3 \). Substitute for \( y \): \[ y = 4(x - 3)^2 - 19(x - 3) + 5 \ = 4(x^2 - 6x + 9) - 19x + 57 + 5 \ = 4x^2 - 24x + 36 - 19x + 57 + 5 \ = 4x^2 - 43x + 98 , (\text{very close, but 98 instead of 97}) \]
Based on this analysis:
- Option 4 is the closest to the given function \( y = 4x^2 - 43x + 97 \), differing by just a constant term. The key points in transforming the original equation to parametric equations were mostly accurate.
Therefore, while none of the options match exactly, Option 4 is the best choice as it matches the form of the quadratic function most closely.