Understanding Polyhedra and Cartesian Coordinates

Topic 11: Polyhedra and Their Geometric Shapes
Polyhedra are geometric shapes with polygonal faces. Euler's Formula: c + v = a + 2. Regular Polyhedra: When all the faces are regular, equal polygons, and moreover, each vertex converges at the same number of faces. Convex: In a convex polyhedron, none of its faces are cut. Concave: These are polyhedra in which at least one of the faces extends inward. Prism: A polyhedron with two parallel faces called bases, and the other faces are parallelograms. The classes of prisms are: straight, regular, irregular, oblique, and parallelepipeds. Area: P_b * h + P_b + apothem / 2 = A_l + 2 * A_b.

Pyramid: A pyramid is a polyhedron with a polygonal base and all other faces are triangles that converge at a point. Area: A_L + A_B => P_B * a / 2 + P_B + a' / 2. Cylinder: A geometric shape generated by rotating a rectangle around one of its sides. Area: 2πrh + 2πr². Area with: πr_generatrix + πr². Area: 4πr².

Topic 13: Cartesian Coordinates: P of a point in the plane is determined by an ordered pair, (x, y), known as Cartesian coordinates. It describes functional relations between two numerical variables, x and y, such that each corresponding value of x has a value of y. The independent variable is x, while the dependent variable depends on x. Continuous Graphs: A continuous graph can be traced without lifting the pencil. Discontinuous Graphs: A discontinuous graph cannot be traced without lifting the pencil. Axes: The points of intersection of the graph with the coordinate axes are called intercepts. The points of intersection with the x-axis are of the form (a, 0). We find these values by calculating the variable when it equals 0. The points of intersection with the y-axis are of the form (0, b). We find these values by calculating the variable when x equals 0.

Increasing and Decreasing Functions: A function is increasing if, as x increases, y also increases. A function is decreasing if, as x increases, y decreases. The maximum point is where the graph transitions from decreasing to increasing, while the minimum point is where it transitions from increasing to decreasing. Proportionality: Directly related functions are directly proportional. y = mx, where m is the constant of proportionality, known as the slope. If m is positive, the function is increasing; if m is negative, the function is decreasing. Inverse Proportionality: Inverse functions relate two variables that are inversely proportional. y = k / x, where k is the constant of proportionality.