The equation of the parabola: definitions and properties

Summary

This lesson defines the parabola as the set of points equidistant from a fixed point, the focus, and from a fixed line, the directrix, and derives its equation from that definition. With the translation of graphs it obtains the equation of a parabola with its vertex at any point and shows that the graphs of quadratic polynomials are exactly the parabolas with a vertical axis: completing the square gives their vertex, their focus and their directrix. It ends with the intersection of parabolas and lines, and with tangent lines.

Learning objectives

State the definition of the parabola by means of the focus and the directrix, and derive its equation from it.
Translate graphs by replacing $x$ with $x - h$ and $y$ with $y - k$ .
Find the vertex, the focus and the directrix of a parabola given by a quadratic polynomial.
Determine the intersection of a parabola with a line and the tangent lines with a given slope.

Two preliminary ideas

The distance between two points

The definition of the parabola rests on distances. As was seen in the lesson on the equation of a line, the distance between two points of the plane is obtained with the Pythagorean theorem:

d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

The distance from a point $(x, y)$ to a horizontal line $y = c$ is the length of the vertical line segment joining them, that is, the difference between $y$ and $c$ taken with a positive sign. It is convenient to write it as the square root of the square of that difference, which is always positive or zero:

\sqrt{(y - c)^2}

The translation of graphs

The theorem on the transformations of a graph, proved in the lesson on function operations, states that the graph of $y = f(x - h) + k$ is that of $y = f(x)$ shifted $h$ units to the right and $k$ upwards. Its proof does not use the fact that the curve is the graph of a function: it holds for any equation between $x$ and $y$ , because the point $(a, b)$ satisfies the original equation exactly when the point $(a + h, b + k)$ satisfies the one that results from replacing $x$ with $x - h$ and $y$ with $y - k$ . This is how the equations of parabolas will be translated.

ExampleTranslating a parabola

Start from the graph of $y = x^2$ and shift it one unit to the right and one unit upwards.

Proof

$y = x^2$
The original graph has its lowest point at the origin.
$y = (x - \resaltar{1})^2$
We replace $x$ with $x - 1$ : each point moves one unit to the right, and the lowest point moves to $(1, 0)$ .
$y - \resaltar{1} = (x - 1)^2$
We also replace $y$ with $y - 1$ : the graph rises by one unit and the lowest point moves to $(1, 1)$ .