Parabolas are U-shaped curves with unique properties that make them useful in math and real life. They're defined by their vertex, , and , which determine their shape and orientation. Understanding these elements is key to graphing and working with parabolas.
Parabolas show up in surprising places, from the path of a thrown ball to the design of satellite dishes. Their special reflective property makes them ideal for focusing light or signals. Knowing how to work with equations lets us model and solve real-world problems.
Parabola Fundamentals
Graphing parabolas
Parabola symmetrical U-shaped curve with passing through vertex dividing it into two equal parts
Vertex turning point of parabola either minimum (opens upward) or maximum point (opens downward)
Parabolas with vertex at origin (0,0) have equation in standard form y=ax2 where a=0
a>0 parabola opens upward
a<0 parabola opens downward
Parabolas with vertex away from origin (h,k) have equation in vertex form y=a(x−h)2+k where a=0
a determines direction (upward or downward) and width of parabola (larger ∣a∣ narrower parabola)
(h,k) coordinates of vertex
Writing equation of parabola in standard form given vertex and point:
Identify vertex (h,k) and point (x,y) on parabola
Substitute vertex coordinates into standard form y=a(x−h)2+k
Substitute point (x,y) into equation solve for a
Substitute value of a back into standard form to complete equation
Key features of parabolas
Vertex turning point of parabola minimum (opens upward) or maximum point (opens downward)
Focus fixed point inside parabola helps define shape
Distance from vertex to focus 4a1 where a coefficient of x2 in standard form
For parabolas with vertex (h,k) focus located at (h,k+4a1) if opens upward or (h,k−4a1) if opens downward
Directrix horizontal line helps define shape of parabola
Perpendicular distance from vertex to directrix also 4a1
For parabolas with vertex (h,k) equation of directrix y=k−4a1 if opens upward or y=k+4a1 if opens downward
Parabola set of all points in plane equidistant from focus and directrix
chord passing through focus perpendicular to axis of symmetry
Eccentricity measure of how much parabola deviates from circular shape (always 1 for parabolas)
Additional parabola properties
Focal chord any chord passing through focus of parabola
Parabolic mirror reflects all incoming rays parallel to axis of symmetry to focus
Reflective property of parabolas explains their use in various applications (satellite dishes, telescopes)
Applications of Parabolas
Real-world applications of parabolas
Projectile motion path of object launched into air at angle neglecting air resistance
Trajectory follows parabolic path
Equation of path y=−2v02cos2θgx2+tanθx+h where g acceleration due to gravity, v0 initial velocity, θ angle of launch, h initial height
Architectural designs parabolic arches (St. Louis Gateway Arch) and bridges evenly distribute weight minimize stress on structure
Satellite dishes and reflecting telescopes parabolic shape focuses incoming signals or light to single point (focus)
Headlights and flashlights parabolic reflectors direct light rays into parallel beam maximizing illumination in specific direction (spotlights, searchlights)
Key Terms to Review (10)
Axis of symmetry: The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. It passes through the vertex of the parabola, and its equation is $x = h$ where $h$ is the x-coordinate of the vertex.
Coordinate plane: A coordinate plane is a two-dimensional surface on which points are plotted and located by their coordinates. It consists of an x-axis (horizontal) and a y-axis (vertical) that intersect at the origin (0,0).
Directrix: A directrix is a fixed line used in the definition of a conic section. In the context of parabolas, it is used to define the set of points equidistant from a given point (focus) and the line (directrix).
Distance formula: The distance formula calculates the distance between two points in a Cartesian coordinate system. It is derived from the Pythagorean theorem and expressed as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
Ellipse: An ellipse is a set of all points in a plane such that the sum of their distances from two fixed points (foci) is constant. It has an oval shape and can be represented by its standard equation.
Focus: In the context of conic sections, a focus is one of the points used to define and construct an ellipse or a parabola. The distance from any point on the conic section to the focus has specific geometric properties.
Hyperbola: A hyperbola is a type of conic section formed by the intersection of a plane and a double-napped cone. It consists of two disconnected curves called branches.
Johnson: A parabola is a symmetric, U-shaped curve defined as the set of all points in a plane equidistant from a given point called the focus and a given line called the directrix. The standard form of its equation is $y = ax^2 + bx + c$.
Latus rectum: The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry that passes through the focus. It has endpoints on the parabola itself.
Parabola: A parabola is a symmetric curve formed by the set of all points in a plane that are equidistant from a given point called the focus and a given line called the directrix. It is commonly represented by the quadratic function $y = ax^2 + bx + c$.