1. **State the problem:** Given the mapping from $x$ to $y$ values: $$\begin{array}{c|cccccc} x & -2 & -1 & 0 & 1 & 3 & 4 \\ y & 4 & -1 & 1 & 5 & 7 & 9 \\ \end{array}$$ and the last $y$ value is 21 for some $x$, find the value of $x$ corresponding to $y=21$. 2. **Identify the pattern or rule:** Observe the $x$ and $y$ pairs to find a relationship. 3. **Check differences in $x$ and $y$:** - From $x=3$ to $x=4$, $y$ goes from 7 to 9 (increase by 2). - From $x=4$ to unknown $x$, $y$ goes from 9 to 21 (increase by 12). 4. **Try to find a formula for $y$ in terms of $x$:** Check if $y$ fits a linear function $y = mx + b$. Using points $(3,7)$ and $(4,9)$: $$m = \frac{9 - 7}{4 - 3} = 2$$ $$b = y - mx = 7 - 2 \times 3 = 7 - 6 = 1$$ So, $y = 2x + 1$. 5. **Verify with other points:** - For $x=1$, $y=2(1)+1=3$ but given $y=5$, so linear doesn't fit all points. 6. **Try quadratic form:** Assume $y = ax^2 + bx + c$. Use points: - $x=-2$, $y=4$: $4 = 4a - 2b + c$ - $x=0$, $y=1$: $1 = c$ - $x=1$, $y=5$: $5 = a + b + c$ From $x=0$, $c=1$. Substitute $c=1$: - $4 = 4a - 2b + 1 \Rightarrow 4a - 2b = 3$ - $5 = a + b + 1 \Rightarrow a + b = 4$ From $a + b = 4$, $b = 4 - a$. Substitute into $4a - 2b = 3$: $$4a - 2(4 - a) = 3$$ $$4a - 8 + 2a = 3$$ $$6a - 8 = 3$$ $$6a = 11$$ $$a = \frac{11}{6}$$ Then, $$b = 4 - \frac{11}{6} = \frac{24}{6} - \frac{11}{6} = \frac{13}{6}$$ 7. **Final quadratic formula:** $$y = \frac{11}{6}x^2 + \frac{13}{6}x + 1$$ 8. **Find $x$ when $y=21$:** $$21 = \frac{11}{6}x^2 + \frac{13}{6}x + 1$$ Multiply both sides by 6: $$126 = 11x^2 + 13x + 6$$ Bring all terms to one side: $$11x^2 + 13x + 6 - 126 = 0$$ $$11x^2 + 13x - 120 = 0$$ 9. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-13 \pm \sqrt{13^2 - 4 \times 11 \times (-120)}}{2 \times 11}$$ Calculate discriminant: $$13^2 - 4 \times 11 \times (-120) = 169 + 5280 = 5449$$ So, $$x = \frac{-13 \pm \sqrt{5449}}{22}$$ Approximate $\sqrt{5449} \approx 73.82$. Thus, $$x_1 = \frac{-13 + 73.82}{22} \approx \frac{60.82}{22} \approx 2.765$$ $$x_2 = \frac{-13 - 73.82}{22} \approx \frac{-86.82}{22} \approx -3.946$$ 10. **Check which $x$ fits the pattern:** Given the $x$ values are integers or close, $x=3$ is already mapped to $y=7$, so $x \approx 2.765$ is close to 3 but not exact. $x \approx -3.946$ is close to -4. Since the mapping shows $x$ values increasing, the next $x$ after 4 could be 6 or so, but given the problem, the closest integer $x$ for $y=21$ is $x=6$. Check $y$ at $x=6$: $$y = \frac{11}{6} \times 36 + \frac{13}{6} \times 6 + 1 = 66 + 13 + 1 = 80$$ Not 21. Therefore, the exact $x$ value for $y=21$ is approximately $x = \frac{-13 + \sqrt{5449}}{22} \approx 2.765$. **Final answer:** $$x = \frac{-13 + \sqrt{5449}}{22}$$