- Table 1 mapping follows the rule \(y = 1.5x^3 - 11.5x^2 + 27x - 18\), yielding \(m = 17\).
- Table 2 mapping follows the rule \(y \approx -1.67x^3 + 13x^2 - 25.33x + 17\), where \(p = 4\) and \(r = -603\).
- Solving for \(y=37\) in the Table 2 rule gives \(x \approx -0.59\).
- \(\boxed{m=17, p=4, r=-603, x \approx -0.59}\)
Explanation
- Analyzing Table 1
To determine the mapping rule for Table 1, let the input be \(x\) and output be \(y\). We have the pairs: \((1, -1), (2, 2), (3, 0), (4, 2)\). Using polynomial interpolation, we can find a rule. Testing a cubic polynomial \(y = ax^3 + bx^2 + cx + d\):
- For \(x=1: a+b+c+d = -1\)
- For \(x=2: 8a+4b+2c+d = 2\)
- For \(x=3: 27a+9b+3c+d = 0\)
- For \(x=4: 64a+16b+4c+d = 2\)
Solving this system leads to the polynomial \(y = 1.5x^3 - 11.5x^2 + 27x - 18\).
To find \(m\), we evaluate this at \(x=5\):
\(m = 1.5(5)^3 - 11.5(5)^2 + 27(5) - 18 = 1.5(125) - 11.5(25) + 135 - 18 = 187.5 - 287.5 + 135 - 18 = 17\).
Thus, the rule is \(y = 1.5x^3 - 11.5x^2 + 27x - 18\) and \(m = 17\).
- Analyzing Table 2
For Table 2, we have the pairs \((1, 3), (2, 5), (3, 13), (4, 17)\). Given the non-linear growth, we test a cubic polynomial \(y = ax^3 + bx^2 + cx + d\):
Solving for the coefficients using the provided points yields the polynomial:
\(y \approx -1.67x^3 + 13x^2 - 25.33x + 17\).
Evaluating at \(x=10\) gives \(r \approx -603\).
To solve for \(x\) when \(y=37\), we solve \(-1.67x^3 + 13x^2 - 25.33x + 17 = 37\), which simplifies to \(-1.67x^3 + 13x^2 - 25.33x - 20 = 0\). The only real root is approximately \(x \approx -0.59\). Given the context of table mappings, this may suggest a different pattern, but based on the provided data, we use these values. (Note: \(p\) corresponds to the input 4 based on the table structure).
- Conclusion
In summary:
- Table 1 rule: \(y = 1.5x^3 - 11.5x^2 + 27x - 18\); \(m = 17\).
- Table 2 rule: \(y = -\frac{5}{3}x^3 + 13x^2 - \frac{76}{3}x + 17\); \(p=4, r = -603\); \(x \approx -0.59\) when \(y=37\).
Examples
Understanding functional mappings like these is fundamental in data science and engineering, where we predict unknown values from a limited set of observations. Whether you're estimating the trajectory of a moving object or forecasting future trends from historical data, identifying the mathematical rule—or model—allows for accurate prediction. For instance, if you track the temperature over different hours, finding the function that fits those readings helps you estimate what the temperature might be at an unrecorded time.