1. **Stating the problem:** We are given a graph with points (0, -4), (1, -6), and (2, 9) and asked to identify the type of sequence or function it represents and find an explicit formula matching the graph. 2. **Analyzing the options:** - An arithmetic sequence has a constant difference between terms. - A geometric sequence has a constant ratio between terms. - A linear function has the form $y = mx + b$ with constant slope. - An exponential function has the form $y = ab^x$ where $a$ and $b$ are constants and $b > 0$. 3. **Checking the differences:** - Difference from $x=0$ to $x=1$: $-6 - (-4) = -2$ - Difference from $x=1$ to $x=2$: $9 - (-6) = 15$ Since the differences are not constant, it is not arithmetic or linear. 4. **Checking the ratios:** - Ratio from $x=0$ to $x=1$: $\frac{-6}{-4} = 1.5$ - Ratio from $x=1$ to $x=2$: $\frac{9}{-6} = -1.5$ Ratios are not constant and also change sign, so it is not geometric. 5. **Conclusion on type:** The graph is non-linear and does not fit arithmetic, geometric, or linear models. The sharp curve suggests an exponential function or another nonlinear function. 6. **Finding an explicit formula:** Assume an exponential function $y = ab^x$. Using point $(0, -4)$: $$ -4 = ab^0 = a \implies a = -4 $$ Using point $(1, -6)$: $$ -6 = -4b \implies b = \frac{-6}{-4} = 1.5 $$ Check point $(2, 9)$: $$ y = -4(1.5)^2 = -4 \times 2.25 = -9 $$ But the actual $y$ at $x=2$ is 9, not -9, so the exponential model $y = -4(1.5)^x$ does not fit all points. 7. **Alternative approach:** The graph is not a simple exponential function with positive base. The sign change and sharp rise suggest a more complex function or a piecewise function. 8. **Final answer:** The graph cannot be accurately described as arithmetic, geometric, or linear. It is best described as a nonlinear function, possibly exponential-like but not a standard exponential function. **Summary:** The graph represents a nonlinear function, not arithmetic, geometric, or linear. The explicit formula cannot be a simple exponential $y=ab^x$ with constant $a,b$ matching all points. **Slug:** nonlinear function **Subject:** algebra