# Symmetry, foci and directrices for standard parabolas

- explain the role of the axis and reflective property of the parabola that is important in many applications
- examine the focus/directrix definition of a parabola using Cartesian coordinates.

## The standard parabola and quadratic function

The most familiar parabola in Cartesian coordinates has equation \(\normalsize{y=x^2}\), which corresponds to the function \[\Large{f(x)=x^2}.\]It passes through the origin \(\normalsize{[0,0]}\), as well as the points \(\normalsize{[1,1]}\), \(\normalsize{[-1,1]}\), \(\normalsize{[2,4]}\), \(\normalsize{[-2,4]}\) and so on.We observe that this parabola is symmetric: the \(\normalsize{y}\)-axis is a line of symmetry, in other words the reflection \(\normalsize{[x,y]}\) goes to \(\normalsize{[-x,y]}\) takes one point on the parabola to another. This line is the**axis**of the parabola. The point where the axis meets the parabola is the

**vertex**, and this is clearly the origin \(\normalsize{[0,0]}\).Let’s also observe that for small values of \(\normalsize{x}\) close to \(\normalsize{0}\), the \(\normalsize{y}\) values of a point \(\normalsize{[x,y]}\) on the parabola are even much smaller, so the parabola hugs the \(\normalsize{x}\)-axis. We say that the \(\normalsize{x}\)-axis is the

**tangent line**to the parabola at the origin \(\normalsize{[0,0}\).

Q1(E): If \(\normalsize{x=0.001}\) then what is the value of \(\normalsize{y=f(x)=x^2}\)? How about if \(\normalsize{x=-0.00002}\)?

Q2(E): If \(\normalsize{x=10000}\) then what is \(\normalsize{y}\)? How about if \(\normalsize{x=2,000,000}\)?

## Zooming out for a better look

Q3(M): In that figure, can you locate the \(\normalsize{x}\)- and \(\normalsize{y}\)-axes? Can you still see the same points that the parabola passes through in this perspective image? The parabola appears to us as an ellipse. Would this have been a surprise to Apollonius?

## The axis and the reflective property of the parabola

^{© “Lovell telescope” by Delusion23/Wikimedia Commons CC BY SA}

## Definition of a parabola in terms of focus and directrix

*locus*(a somewhat old fashioned word meaning

*path*, or

*trajectory*) of a point \(\normalsize{X}\) whose distance from a fixed point \(\normalsize{F}\) equals its distance from a fixed line \(\normalsize{l}\). Using the rational concept of quadrance, whose square root is the distance, it is algebraically simpler to state this condition in the form

**focus**of the parabola that figured in the reflective property above!! The line \(\normalsize{l}\) is called the

**directrix**. In fact the ellipse and hyperbola have a similar description, as we shall see in a while.

## A calculation

## The focus and directrix of \(\normalsize{y=x^2}\)

Q4(C): See if you can verify this algebraically, following our computation above.

## Answers

A1.If \(\normalsize{x=0.001}\) then \(\normalsize{y=(0.001)^2=0.000001}\). And if \(\normalsize{x=-0.00002}\) then \(\normalsize{y=(-0.00002)^2=0.0000000004=4 \times 10^{-10}}\).A2If \(\normalsize{x=10000}\) then \(\normalsize{y=(10000)^2=100000000=10^8}\). And if \(\normalsize{x=2,000,000}\) then \(\normalsize{y=(2,000,000)^2=(2 \times 10^6)^2=4 \times 10^{12}}\).A3.The \(\normalsize{x}\)- and \(\normalsize{y}\)-axes are marked by darker black lines. Apollonius would not have been surprised. The cone of light from our eye to the parabola is indeed just a cone like he studied, and its intersection with the horizontal plane which we are calling the \(\normalsize{x-y}\) plane is, in this case, an ellipse.A4.We hope your computation succeeded. If not, have another look at our computation and follow it carefully.

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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