Whether in the context of quadratic functions, parabolic mirrors or alternative energy designs like solar cookers, the parabola holds a special place in science and mathematics — particularly geometry.
Learn the parabola equation and its many versatile applications.
What Is a Parabola?
A parabola is a symmetrical, U-shaped curve. It is a type of conic section, a geometric shape that forms through the intersection of a plane and a right circular cone.
Expressing a Parabola in Standard Form
The standard equation for a parabola is:
y = ax² + bx + cYou may recognize this as a quadratic function; a parabola is a graphic depiction of a quadratic function.
In the equation above, a, b and c are constants, and a is not equal to zero. (If a were zero, you would have a linear equation.) Through this equation, you can identify the vertex, axis of symmetry and the direction in which the parabola opens.
Example With Step-by-Step Instructions
Follow these steps to draw a parabola with the given equation: y = 2 2 − 4x + 1
In this case, a is 2, b is -4 and c is 1.
The parabola's vertex is (h,k). This is about to get really confusing, so hang in there with us: h is the x-coordinate of the vertex, and k is the vertex's y-coordinate. Keep that in mind for the rest of this step.
Now, h = -b/2a. Plug in the values into the formula, and you get:
h = -(-4)/(2)(2)This means the x-coordinate of the vertex is 1. Now, find k, the y-coordinate of the vertex, by substituting h back into the original expression (ax 2 + bx + c):
y = 2(1)² – 4(1) + 1 y = 2 – 4 + 1This makes the vertex (1, -1).
The parabola's axis of symmetry (x = h) is a vertical line passing through the vertex, so as demonstrated above, this is x = 1.
You can solve it using the quadratic formula: x = -b ± √(b² – 4ac) / (2a). Substitute the values of a, b and c into the formula.
x = -(-4) ± √((-4)² – 4(2)(1)) / 2(2) x = 4 ± √(16 – 8) / 4 x = 4 ± √8 / 4 x = 4 ± (2√2) / 4Then you calculate the two x-intercepts: one for a + and one for a –.
x = 4 + (2√2) / 4 = 4 + (√2)/2 x = 4 – (2√2) / 4 = 4 + (√2)/2So the x-intercepts are: 2 + (√2)/2 and 2 – (√2)/2.
To find the y-intercept, set x to 0 and solve for y.
y = 2(0)² – 4(0) + 1This makes the y-intercept point is (0,1).
7 Components of a Parabola
From the focal point to the fixed, straight line of the directrix, these are the parabola components that define the shape and properties of the curve.
The vertex is the parabola's minimum or maximum value. It serves as the focal point for both the axis of symmetry and the parabola's overall curvature.
The axis of symmetry, parallel to the y-axis, is an imaginary line passing through the vertex, dividing the parabola into two symmetric halves.
Intercepts provide valuable information about the behavior and characteristics of the parabola. The x-intercepts reveal where the curve crosses the x-axis, and the y-intercept indicates point it intersects the y-axis.
Graphically, these intercepts are points on the curve that help define its shape and position in the coordinate plane.
A parabola opens upward or downward. The coefficient a in the quadratic equation determines the direction in which the parabola opens. If the coefficient is positive, it opens upward; if negative, the parabola opens downward.
In the context of conic sections, the focus is a fixed point through which all light rays parallel to the axis of symmetry will reflect off the parabola. The directrix is a fixed line perpendicular to the axis of symmetry.
The latus rectum, or focal chord, is a line segment that passes through the focus of the parabola and is perpendicular to the axis of symmetry.
The focal distance of a parabola is the distance from the vertex (the highest or lowest point on the parabola) to the focus.