1. **Problem Statement:** We want to recreate the train image using equations of geometric shapes on a coordinate plane. 2. **Shapes and their equations:** - **Circles:** Use the equation $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h,k)$ is the center and $r$ is the radius. - **Rectangles:** Use inequalities for $x$ and $y$ to define boundaries, e.g., $$x_1 \leq x \leq x_2, \quad y_1 \leq y \leq y_2$$. - **Trapezoid (funnel):** Use linear equations for the four sides connecting the vertices. - **Line (tracks):** Use $y = c$ for horizontal lines and vertical lines at specific $x$ values. 3. **Equations for the train components:** - **Smoke circles:** - Circle 1: $$(x + 2)^2 + (y - 2)^2 = r_1^2$$ - Circle 2: $$(x + 1.4)^2 + (y - 2.3)^2 = r_2^2$$ - Circle 3: $$(x + 0.8)^2 + (y - 2.5)^2 = r_3^2$$ Choose radii $r_1, r_2, r_3$ around 0.3 to 0.4. - **Funnel (trapezoid):** Vertices approximately at $(-3.5, 0.5), (-2.6, 0.5), (-3.2, 1.5), (-2.7, 1.5)$. Use line equations for each side: - Bottom: $y=0.5$ for $-3.5 \leq x \leq -2.6$ - Top: $y=1.5$ for $-3.2 \leq x \leq -2.7$ - Left side: line through $(-3.5,0.5)$ and $(-3.2,1.5)$ - Right side: line through $(-2.6,0.5)$ and $(-2.7,1.5)$ - **Headlamp (circle):** Center $(-3.5, 0)$, radius about 0.3. - **Main engine rectangles:** - Front rectangle: $$-3.8 \leq x \leq -2.8, \quad -1 \leq y \leq 1$$ - Middle cab rectangle: $$-1.5 \leq x \leq -1, \quad 0.5 \leq y \leq 1.5$$ - Rear rectangle: $$0 \leq x \leq 1, \quad -1 \leq y \leq 1$$ - **Wheels (circles):** - Front wheel: center $(-3, -2)$, radius about 0.5 - Middle wheel: center $(-1, -2)$, radius about 0.8 - Two smaller wheels: centers $(2, -2)$ and $(4, -2)$, radius about 0.5 - **Train car rectangle:** $$1 \leq x \leq 5, \quad -1 \leq y \leq 1$$ - **Windows (circles):** - Window 1: center $(2, 0)$, radius about 0.5 - Window 2: center $(3.5, 0)$, radius about 0.5 - **Trees:** - Left tree circle: center $(-4, -1)$, radius about 0.5 - Right tree circle: center $(5, -1)$, radius about 0.5 - Tree trunks: vertical lines below each tree circle - **Tracks:** - Horizontal line: $$y = -3$$ - Vertical ticks: short vertical lines at intervals beneath wheels, e.g., at $x = -3, -1, 2, 4$ 4. **Summary:** Using these equations and inequalities, you can plot each shape to recreate the train image on the coordinate plane. This approach uses basic geometry equations to model each component precisely. Final note: Adjust radii and exact coordinates slightly for best visual match.