In order to sketch a parabola, we need to first mark the focus of the parabola at (0,0). Then, we will draw a horizontal line, which is a horizontal directrix of the parabola, 6 units below the focus.
Since the focus and directrix are equidistant from every point on the on the parabola, the cricles on graph paper help us count the distance between points and draw the parabola. Since the directrix is 6 units below the focus, let's begin by finding the point 6 ÷ 2 = 3 units away from both focus and the directrix.
Using the information that horizontal dirctrix is 6 units below the focus, let's locate points that are 6 units away from both focus and the directrix. They will lie on the circle that has radius equal to 6.
To find another points that are equidistant from both focus and directrix, we have to first choose the distance we are interested in. This time let it be equal to 13. We will find the point that lies the vertical dashed line perpendicular to the horizontal directrix and on the circle with radius 13.
We can sketch the parabola by connecting the obtained points with a smooth line.
We can move on to writing an expression for the distance from the focus to any point on parabola. Let's recall the distance formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Let (x,y) be a point on the parabola. We will find an expression for the distance between (x,y) and the focus (0,0). To do so, we will substitute (x,y) and (0,0) for (x_1,y_1) and (x_2,y_2), respectively, in the above formula.
b We will also use the distance formula to find an expression for the distance from the directrix to any point (x,y) on the parabola. The distance between a point and a line is through a perpendicular segment.
Therefore, the distance between the point and the directrix is the same as the distance between (x,y) and (x, - 6).
