Written out as an equation, the formula looks like: F + V - E = 2 F refers to the number of faces V refers to the number of vertices, or corner points E refers to the number of edges
F refers to the number of faces V refers to the number of vertices, or corner points E refers to the number of edges
V = 2 - F + E
Example: For a polyhedron that has 6 faces and 12 edges. . . V = 2 - F + E V = 2 - 6 + 12 V = -4 + 12 V = 8
If using a graphing calculator to graph the inequalities, you can usually scroll over to the vertices and find the coordinates that way.
Example: For the system of inequalities: y < x y > -x + 4 Change the inequalities to: y = x y = -x + 4
Example: For the system of inequalities: y < x y > -x + 4 Change the inequalities to: y = x y = -x + 4
Example: For the system of inequalities: y < x y > -x + 4 Change the inequalities to: y = x y = -x + 4
Example: If: y = x y = -x + 4 Then y = -x + 4 can be written as: x = -x + 4
Example: If: y = x y = -x + 4 Then y = -x + 4 can be written as: x = -x + 4
Example: If: y = x y = -x + 4 Then y = -x + 4 can be written as: x = -x + 4
Example: If: y = x y = -x + 4 Then y = -x + 4 can be written as: x = -x + 4
Example: If: y = x y = -x + 4 Then y = -x + 4 can be written as: x = -x + 4
Example: If: y = x y = -x + 4 Then y = -x + 4 can be written as: x = -x + 4
x = -x + 4
Example: x = -x + 4 x + x = -x + x + 4 2x = 4 2x / 2 = 4 / 2 x = 2
Example: x = -x + 4 x + x = -x + x + 4 2x = 4 2x / 2 = 4 / 2 x = 2
Example: y = x y = 2
Example: y = x y = 2
Example: y = x y = 2
Example: (2, 2)
Example: (2, 2)
Example: (2, 2)
Example: (using decomposition) 3x2 - 6x - 45 Factor out the common factor: 3 (x2 - 2x - 15) Multiply the a and c terms: 1 * -15 = -15 Find two numbers with a product that equals -15 and a sum that equals the b value, -2: 3 * -5 = -15; 3 - 5 = -2 Substitute the two values into the equation ax2 + kx + hx + c: 3(x2 + 3x - 5x - 15) Factor the polynomial by grouping: f(x) = 3 * (x + 3) * (x - 5)
Example: 3 * (x + 3) * (x - 5) = 0 х +3 = 0 х - 5 = 0 х = -3 ; х = 5 Therefore, the roots are: (-3, 0) and (5, 0)
Example: x = 1; this value lies directly between -3 and 5
Example: y = 3x2 - 6x - 45 = 3(1)2 - 6(1) - 45 = -48
Example: (1, -48)
Example: y = -x^2 - 8x - 15
Example: y = -x^2 - 8x - 15
Example: y = -x^2 - 8x - 15
Example: -1 (x^2 + 8x) - 15
Example: 8 / 2 = 4; 4 * 4 = 16; therefore, -1(x^2 + 8x + 16) Also keep in mind that what you do to the inside must also be done to the outside: y = -1(x^2 + 8x + 16) - 15 + 16
Example: y = -1(x + 4)^2 + 1
k = 1 h = -4 Therefore, the vertex of this equation can be found at: (-4, 1)
k = 1 h = -4 Therefore, the vertex of this equation can be found at: (-4, 1)
k = 1 h = -4 Therefore, the vertex of this equation can be found at: (-4, 1)
Example: y = -x^2 - 8x - 15 x = -b / 2a = -(-8)/(2*(-1)) = 8/(-2) = -4 x = -4
Example: y = -x^2 - 8x - 15 x = -b / 2a = -(-8)/(2*(-1)) = 8/(-2) = -4 x = -4
Example: y = -x^2 - 8x - 15 x = -b / 2a = -(-8)/(2*(-1)) = 8/(-2) = -4 x = -4
Example: y = -x^2 - 8x - 15 = -(-4)^2 - 8(-4) - 15 = -(16) - (-32) - 15 = -16 + 32 - 15 = 1 y = 1
Example: (-4, 1)
