Identify the direction a parabola opens

Learning Objectives Identify the orientation (vertical or horizontal) of a parabola by inspecting which variable is squared in its equation. Determine if a vertical parabola opens upwards or downwards from its standard form (y = ax^2 + bx + c) by analyzing the sign of 'a'. Determine if a vertical parabola opens upwards or downwards from its vertex form (y = a(x-h)^2 + k) by analyzing the sign of 'a'. Determine if a parabola opens up, down, left, or right from its conic section form by analyzing the sign of the parameter 'p'. Differentiate between the rules for vertically-opening parabolas ((x-h)^2 = 4p(y-k)) and horizontally-opening parabolas ((y-k)^2 = 4p(x-h)). Rearrange a parabola's equation from general form to a standard form to identify...

TermDefinitionExample Vertical ParabolaA parabola that opens either upwards or downwards, with a vertical axis of symmetry. In its equation, the 'x' variable is squared.y = 2x^2 - 4x + 5 Horizontal ParabolaA parabola that opens either to the left or to the right, with a horizontal axis of symmetry. In its equation, the 'y' variable is squared.x = -y^2 + 3y - 1 Leading Coefficient (a)In the standard or vertex form of a vertical parabola, 'a' is the coefficient of the x^2 term. Its sign determines if the parabola opens up or down.In y = -3x^2 + 7, the leading coefficient 'a' is -3. Conic Section FormAn equation form that highlights a parabola's geometric properties, including its vertex (h,k) and the parameter 'p'. There are separate fo...

Direction Rule for Vertical Parabolas For equations `y = ax^2 + bx + c` or `(x-h)^2 = 4p(y-k)`: 1. If `a > 0` or `p > 0`, the parabola opens UPWARDS. 2. If `a < 0` or `p < 0`, the parabola opens DOWNWARDS. Use this rule when you identify that the 'x' variable is squared. The direction depends on the sign of the coefficient of the squared term ('a') or the sign of the parameter 'p'. Direction Rule for Horizontal Parabolas For equations `x = ay^2 + by + c` or `(y-k)^2 = 4p(x-h)`: 1. If `a > 0` or `p > 0`, the parabola opens to the RIGHT. 2. If `a < 0` or `p < 0`, the parabola opens to the LEFT. Use this rule when you identify that the 'y' variable is squared. The direction depends on the sign of the coefficient...