1. **State the problem:** We are given a point $(1,4)$ on the graph of $y=f(x)$ and need to find the coordinates of its image on the graph of $y=3f[-4(x+1)]-2$. 2. **Understand the transformation:** The new function is $y=3f[-4(x+1)]-2$. This involves several transformations: - Inside the function: $-4(x+1)$ means a horizontal shift left by 1 and a horizontal scaling by factor $\frac{1}{4}$ with reflection (due to the negative sign). - Outside the function: multiplication by 3 scales vertically by 3. - Subtracting 2 shifts the graph down by 2. 3. **Find the new $x$ coordinate:** Let the original $x$ be $x_0=1$. We want to find $x$ such that: $$-4(x+1) = x_0 = 1$$ Solve for $x$: $$-4(x+1) = 1$$ $$x+1 = \frac{1}{-4} = -\frac{1}{4}$$ $$x = -\frac{1}{4} - 1 = -\frac{5}{4}$$ 4. **Find the new $y$ coordinate:** The original $y$ is $f(1) = 4$. The new $y$ is: $$y = 3f[-4(x+1)] - 2 = 3f(1) - 2 = 3 \times 4 - 2 = 12 - 2 = 10$$ 5. **Final answer:** The image of the point $(1,4)$ on the graph of $y=3f[-4(x+1)]-2$ is: $$\boxed{\left(-\frac{5}{4}, 10\right)}$$