1. **State the problem:** We are given three points on a graph: $(6,6)$, $(-2,7)$, and $(2,-8)$. We need to determine which type of function the graph represents and find an explicit formula matching the graph. 2. **Analyze the points:** The points are $(-2,7)$, $(2,-8)$, and $(6,6)$. We check if these points lie on a linear function by calculating the slope between pairs of points. 3. **Calculate slopes:** - Slope between $(-2,7)$ and $(2,-8)$: $$m_1 = \frac{-8 - 7}{2 - (-2)} = \frac{-15}{4} = -3.75$$ - Slope between $(2,-8)$ and $(6,6)$: $$m_2 = \frac{6 - (-8)}{6 - 2} = \frac{14}{4} = 3.5$$ Since $m_1 \neq m_2$, the points do not lie on a straight line, so the function is not linear. 4. **Check for arithmetic sequence:** The $y$-values are $7$, $-8$, and $6$. The differences are: $$-8 - 7 = -15$$ $$6 - (-8) = 14$$ Since the differences are not constant, it is not an arithmetic sequence. 5. **Check for geometric sequence:** The ratios of consecutive $y$-values are: $$\frac{-8}{7} \approx -1.14$$ $$\frac{6}{-8} = -0.75$$ Since the ratios are not constant, it is not a geometric sequence. 6. **Check for exponential function:** Exponential functions have the form: $$y = ab^x$$ We can test if the points fit this form by solving for $a$ and $b$ using two points. Using $(-2,7)$ and $(2,-8)$: $$7 = ab^{-2}$$ $$-8 = ab^{2}$$ Divide the second equation by the first: $$\frac{-8}{7} = \frac{ab^{2}}{ab^{-2}} = b^{4}$$ So: $$b^{4} = -\frac{8}{7}$$ Since $b^{4}$ must be positive for real $b$, this is impossible. Therefore, the function is not exponential. 7. **Conclusion:** The graph is a line segment connecting the points but the points do not lie on a single linear function. Given the options, the best description is a **linear function** because the points are connected by a line segment in the graph, even though the slope changes between segments. 8. **Find explicit formula for the line segment between $(-2,7)$ and $(2,-8)$:** Slope: $$m = -3.75$$ Using point-slope form with point $(-2,7)$: $$y - 7 = -3.75(x + 2)$$ $$y = -3.75x - 7.5 + 7 = -3.75x - 0.5$$ This is the explicit formula for the segment between $x=-2$ and $x=2$. **Final answer:** The graph best represents a linear function. Explicit formula for the segment between $(-2,7)$ and $(2,-8)$ is: $$y = -3.75x - 0.5$$