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This can be done with calculus, or by using a line that is parallel to the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola. [e] Then, using the formula given in Distance from a point to a line, calculate the perpendicular distance from this point to the ...
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
Equivalently, this is the graph of the bivariate quadratic equation = + +. If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards. The coefficient a controls the degree of curvature of the graph; a larger magnitude of a gives the graph a more closed (sharply curved) appearance.
The simplest is the slope-intercept form: = +, from which one can immediately see the slope a and the initial value () =, which is the y-intercept of the graph = (). Given a slope a and one known value () =, we write the point-slope form:
Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation =. The slope at a point of the parabola is m = 2 a x {\displaystyle m=2ax} . Replacing x gives the parametric representation of the parabola with the tangent slope as parameter: ( m 2 a , m 2 4 a ) . {\displaystyle \left({\tfrac {m}{2a ...
Substituting into the formula for general parametrizations gives exactly the same result as above, with x replaced by t. If we use primes for derivatives with respect to the parameter t. The same parabola can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = ax 2 + bx + c – y.
A hyperbola can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two asymptotes of the hyperbola. Likewise, a parabola can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis ...
Consider, for example, the one-parameter family of tangent lines to the parabola y = x 2. These are given by the generating family F(t,(x,y)) = t 2 – 2tx + y. The zero level set F(t 0,(x,y)) = 0 gives the equation of the tangent line to the parabola at the point (t 0,t 0 2).