1. **Stating the problem:** We are asked to perform a complete study of the function represented by the first graph (a) and then use it as an example to analyze the other six graphs. 2. **Understanding the example graph (a):** - The graph is piecewise linear with vertices at (-4,3), (-3,0), (-3,3), (0,3), (2,0), (4,0), (6,1), and a point at (3,-3). - We analyze domain, range, extrema, monotonicity, zeros, and values at specific points. 3. **Step-by-step analysis of graph (a):** **Domain:** The set of all x-values covered by the graph. From the points, domain is approximately $[-4,6]$. **Range:** The set of all y-values. From points, range is approximately $[-3,3]$. **Extrema:** - Maxima: At points where the function changes from increasing to decreasing or has a peak. - Minima: At points where the function changes from decreasing to increasing or has a valley. **Monotonicity:** - Identify intervals where the function is increasing or decreasing by looking at the slope between points. **Zeros:** Points where $f(x)=0$. **Value at specific points:** For example, $f(0)=3$. 4. **Applying the same method to the other graphs:** **Graph b:** - Domain: $[-8,10]$ - Range: $[-4,6]$ - Extrema: Max at (-2,6) and (10,6), Min at (-4,-2) and (2,-4) - Monotonicity: Increasing and decreasing intervals determined by slopes between points - Zeros: At $x=0$ and $x=2$ **Graph c:** - Domain: $[-10,80]$ - Range: $[-20,70]$ - Extrema: Max at (-10,70), Min at (70,-20) - Monotonicity: Decreasing from (-10,70) to (10,10), increasing and decreasing in other intervals - Zeros: At (40,0) and (80,0) **Graph d:** - Domain and range approximately $[0,80]$ - Extrema and monotonicity inferred from shape **Graph e:** - Parabola with vertex at (-5,-26.5) - Domain: All real numbers - Range: $[-26.5, +)$ - Minimum at vertex **Graph f:** - Sinusoidal shape with peaks at (-23,24.6) and troughs at (23,-24.6) - Domain: All real numbers - Range: $[-24.6,24.6]$ 5. **Summary:** For each graph, identify domain, range, extrema, monotonicity, zeros, and special values by analyzing the given points and shape, following the example of graph (a).