1. **State the problem:** We have a function $f(t)$ graphed and three new functions defined as: - $a(t) = f(t) + 3$ - $b(t) = f(t + 4)$ - $c(t) = f(t - 4) - 2$ We need to find the values of: (a) $a(0.9)$ (b) $a(9.4)$ (c) $b(\frac{53}{10})$ (d) $c(9)$ (e) $c(\frac{6}{5}) = -4$ (given) 2. **Recall function transformations:** - Adding a constant outside the function shifts the graph vertically. - Adding a constant inside the function argument shifts the graph horizontally. 3. **Evaluate each value step-by-step:** (a) $a(0.9) = f(0.9) + 3$ From the graph description, for $t$ between $-3$ and $1$, $f(t) = -3$ (flat line). So, $f(0.9) = -3$ Therefore, $a(0.9) = -3 + 3 = 0$ (b) $a(9.4) = f(9.4) + 3$ At $t=9.4$, the graph is descending below zero but above $-3$. Since the exact value is not given, approximate from the description: The graph ends below zero at $t=10$, so at $9.4$ it is slightly above that, say approximately $-1.5$ (estimate). Thus, $a(9.4) \approx -1.5 + 3 = 1.5$ (c) $b(\frac{53}{10}) = b(5.3) = f(5.3 + 4) = f(9.3)$ At $t=9.3$, similar to above, approximate $f(9.3) \approx -1.5$ So, $b(5.3) \approx -1.5$ (d) $c(9) = f(9 - 4) - 2 = f(5) - 2$ At $t=5$, the graph is near the peak at $(6,3)$, so $f(5)$ is slightly less than 3, estimate $f(5) \approx 2.5$ Therefore, $c(9) \approx 2.5 - 2 = 0.5$ (e) Given $c(\frac{6}{5}) = -4$ Check consistency: $c(\frac{6}{5}) = f(\frac{6}{5} - 4) - 2 = f(-2.8) - 2$ Since $f(t) = -3$ for $t$ between $-3$ and $1$, $f(-2.8) = -3$ So, $c(\frac{6}{5}) = -3 - 2 = -5$, but given is $-4$, so there might be a discrepancy or typo. 4. **Final answers:** (a) $0$ (b) $1.5$ (approximate) (c) $-1.5$ (approximate) (d) $0.5$ (approximate) (e) Given as $-4$, but calculation suggests $-5$