Question: Given the function $C(t) = (t - 1)(t + 5)(t - 4)$ Remember that intercepts are points on a graph and must be given in coordinate form. Its C-intercept is Its t-intercept(s) is/are

1. **State the problem:** Find the $C$-intercept and the $t$-intercepts of the function $$C(t) = (t - 1)(t + 5)(t - 4).$$ 2. **Recall definitions:** - The $C$-intercept is the point where the graph crosses the $C$-axis (vertical axis), which occurs when $t=0$. - The $t$-intercepts are the points where the graph crosses the $t$-axis (horizontal axis), which occur when $C(t) = 0$. 3. **Find the $C$-intercept:** Substitute $t=0$ into the function: $$C(0) = (0 - 1)(0 + 5)(0 - 4) = (-1)(5)(-4) = 20.$$ So the $C$-intercept is at the point $(0, 20)$. 4. **Find the $t$-intercepts:** Set $C(t) = 0$: $$(t - 1)(t + 5)(t - 4) = 0.$$ This product is zero if any factor is zero: - $t - 1 = 0 \implies t = 1$ - $t + 5 = 0 \implies t = -5$ - $t - 4 = 0 \implies t = 4$ So the $t$-intercepts are at points $(1, 0)$, $(-5, 0)$, and $(4, 0)$. **Final answer:** - $C$-intercept: $(0, 20)$ - $t$-intercepts: $(1, 0)$, $(-5, 0)$, $(4, 0)$