Parabola: Focus and Directrix

• A parabola is a 2-dimensional U-shaped curve representing the graph of a quadratic equation, which can open upwards or downwards.

Key Features of a Parabola

• Vertex: The maximum point of a downward-facing parabola or the minimum point of an upward-facing parabola.
• Axis of Symmetry: A line passing through the vertex that divides the parabola into two symmetrical halves.
• A parabola consists of all points equidistant from a point called the focus and a line called the directrix.

Finding the Focus of a Parabola

• Vertex form of a parabola: [ y = a(x - h)^2 + k ], where (h, k) is the vertex.
• The focus of a parabola in vertex form is the point ((h, k + p)].
• Positive ( h ) values shift the parabola to the right, and negative ( h ) values shift it to the left.
• Positive ( k ) values shift the parabola upwards, and negative ( k ) values shift it downwards.

Finding the Directrix of a Parabola

• Standard form of a parabola: [ y = a(x - h)^2 + k ], where ( h ) and ( k ) represent shifts, and ( p ) affects the width.
• The directrix for a parabola in this form is the line [ y = k - p ].
• Higher ( p ) values make the parabola narrower, while lower ( p ) values make it wider.
• The directrix is a horizontal line for parabolas opening upwards or downwards.
• Parabolas of the form [ x = ay^2 + by + c ] open to the right or left and can have a vertical directrix.

Completing the Square

• Completing the square is essential for converting a parabola from quadratic equation form to standard form.
• Given a quadratic equation [ ax^2 + bx + c ], factor out ( a ) to get [ a(x^2 + \frac{b}{a}x) + c ].
• Add and subtract [ (\frac{b}{2a})^2 ] inside the parentheses to complete the square.
• Convert the equation into the form [ a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a}) ].

Finding the Equation of a Parabola

• Given the directrix and focus, determine the equation of the parabola.
• For a parabola in standard form [ y = a(x - h)^2 + k ], the directrix is [ y = k - p ] and the focus is ((h, k + p)].
• Use the relationships [ k - p = m ] (directrix) and [ k + p = n ] (focus) to solve for ( p ), ( h ), and ( k ).
• The equation of the parabola in standard form can then be determined.
• Vertex form can simply be converted to standard form using algebra.

Example 1

• Given parabola: [ y = 4x^2 + 1 ].
• Convert to standard form to find [ p = \frac{1}{16} ].
• Focus: ((0, \frac{17}{16})].
• Directrix: [ y = \frac{15}{16} ].The parabola opens upwards.

Example 2

• Given parabola: [ x = (y - 4)^2 - 4 ].
• The parabola opens to the right, with [ h = -4 ], [ k = 4 ], and [ p = 1 ].
• Directrix: [ x = 3 ].
• Focus: ((5, -4)].

Example 3

• Given vertex: ((0, 1)) and focus: ((0, -1)).
• Determine [ h = 0 ], [ k = 1 ], and [ p = -2 ].
• Directrix: [ y = 3 ].
• Standard form of the parabola: [ y = -\frac{1}{8}x^2 + 1 ]. The parabola opens downwards.