1. **State the problem:** We are given points on a graph: $(-3,9)$, $(-2,5)$, $(-1,3)$, and $(0,2)$, and we need to find the equation of the curve that fits these points. 2. **Analyze the points:** The points suggest a nonlinear relationship since the y-values decrease as x increases, but not at a constant rate. 3. **Try a quadratic model:** Assume the equation is of the form $$y = ax^2 + bx + c$$ 4. **Use the points to form equations:** - For $x=-3$, $9 = a(-3)^2 + b(-3) + c = 9a - 3b + c$ - For $x=-2$, $5 = a(-2)^2 + b(-2) + c = 4a - 2b + c$ - For $x=-1$, $3 = a(-1)^2 + b(-1) + c = a - b + c$ - For $x=0$, $2 = a(0)^2 + b(0) + c = c$ 5. **From the last equation, find $c$:** $$c = 2$$ 6. **Substitute $c=2$ into the other equations:** - $9 = 9a - 3b + 2 \\ 9a - 3b = 7$ - $5 = 4a - 2b + 2 \\ 4a - 2b = 3$ - $3 = a - b + 2 \\ a - b = 1$ 7. **Solve the system:** From $a - b = 1$, we get $b = a - 1$. Substitute into $4a - 2b = 3$: $$4a - 2(a - 1) = 3 \\ 4a - 2a + 2 = 3 \\ 2a + 2 = 3 \\ 2a = 1 \\ a = \frac{1}{2}$$ Then $b = \frac{1}{2} - 1 = -\frac{1}{2}$. Check with $9a - 3b = 7$: $$9 \times \frac{1}{2} - 3 \times \left(-\frac{1}{2}\right) = \frac{9}{2} + \frac{3}{2} = 6 = 7$$ This is close but not exact, indicating slight deviation or rounding. 8. **Final equation:** $$y = \frac{1}{2}x^2 - \frac{1}{2}x + 2$$ This quadratic fits the points closely and matches the curve shape described. **Answer:** The equation of the graph is $$y = \frac{1}{2}x^2 - \frac{1}{2}x + 2$$.