1. **Problem Statement:** Given points (-3, -5), (1, -6), and (5, -7) on a graph, determine the type of function they represent and find an explicit formula for the function. 2. **Identify the function type:** The points lie on a straight line with a constant rate of change. This suggests a linear function of the form $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Calculate the slope $m$:** Use the formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using points (-3, -5) and (1, -6): $$m = \frac{-6 - (-5)}{1 - (-3)} = \frac{-6 + 5}{1 + 3} = \frac{-1}{4} = -\frac{1}{4}$$ 4. **Find the y-intercept $b$:** Use point-slope form with point (1, -6): $$y = mx + b \Rightarrow -6 = -\frac{1}{4} \times 1 + b$$ $$b = -6 + \frac{1}{4} = -\frac{24}{4} + \frac{1}{4} = -\frac{23}{4}$$ 5. **Write the explicit formula:** $$y = -\frac{1}{4}x - \frac{23}{4}$$ 6. **Interpretation:** The function is a linear function with a negative slope, decreasing by 1 unit in $y$ for every 4 units increase in $x$. This matches the description of the graph. **Final answer:** $$y = -\frac{1}{4}x - \frac{23}{4}$$