Find the focus and directrix of the parabola with the given equation y2 = 16x. The directrix is given by the equation. fe the solutions are imaginary numbers, the parabola doesn't intersect the x axis. Notice that this doesn't give us any information about the location of the focus or directrix. parabola. As the angle A in the animation below changes, it eventually becomes equal to B and the conic section is a parabola. (we can deduce this from simple calculus), If ɑ is positive, the parabola will open up, If ɑ is negative the parabola will open down, Plug the value -b/2ɑ into the equation to get the value of y, The coefficient a is positive, so the parabola opens up and the vertex is a minimum, ɑ = 5 and b = -10 so the minimum occurs at -b/2ɑ = - (-10)/(2(5)) = 1, So the x axis intercepts occur at (-2, 0) and (-1/3, 0), The equation of the parabola in focus vertex form is (x - h), The vertex is at (h,k) giving us h = 4, k = 6, The focus is located at (h, k + p).

Find the vertex, focus and directrix. A and B are the x-intercepts of the parabola y = ax² + bx + c and roots of the quadratic equation ax² + bx + c = 0, Example 1: Find the x-axis intercepts of the parabola y = 3x2 + 7x + 2, Example 1: Find the x-intercepts of the parabola y = 3x2 + 7x + 2, Example 2: Find the x-axis intercepts of the parabola with vertex located at (4, 6) and focus at (4, 3), Example 2: Find the x-intercepts of the parabola with vertex at (4, 6) and focus at (4, 3). The purple lines in the picture below represent the distance between the focus and different points on the directrix .

if ɑ = 1 and (h,k) is the origin (0,0) we get the simple parabola we saw at the start of the tutorial: Vertex form of the equation of a parabola. Conversely in the case of a headlight or torch, light coming from the focus will be reflected off the reflector and travel outwards in a parallel beam. But k = 6 so p = 3 - 6 = -3, Plug the values into the equation (x - h), The coefficients are a = -1/12, b = 2/3, c = 14/3, This gives us x = -4.49 approx and x = 12.49 approx, So the x axis intercepts occur at (-4.49, 0) and (12.49, 0), When you kick a ball into the air or a projectile is fired, the trajectory is a parabola, The reflectors of vehicle headlights or flashlights are parabolic shaped, The mirror in a reflecting telescope is parabolic, Satellite dishes are in the shape of a parabola as are radar dishes. We can rewrite this as x2 = y, but the coefficient of y is 1, so 4p must equal 1 and p = 1/4. Also covered will be how to work out the maxima and minima of a parabola and how to find the intersection with the x and y axes. In the equations, ɑ is a coefficient and can have any value.

In the next section, we will explain how the focus and directrix relate to the actual parabola. A parabola is the shape produced when a plane intersects a cone and the angle of intersection to the axis is equal to half the opening angle of the cone.

A parabola is a locus of points equidistant (the same distance) from a line called the directrix and point called the focus. In this example the focus is at (4, 3) so k + p = 3. Open down means it will have a maximum and the value of y decreases on both sides of the max. Parabolas with different coefficients of x².


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