Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Parabolas

The Graph of $ y=x^2 $.


Figure 1. A parabola

The graph of an equation involving $ x $ and $ y $ is the set of all points $ (x,y) $ in the cartesian coordinate plane whose coordinates satisfy the equation. The graph of a function $ f $ is the graph of the equation $ y=f(x). $ For example, Figure 1 shows the graph of the equation

$\displaystyle y= x^2. $

This particular graph is an example of a parabola. Note that it is symmetric with respect to the $ y $-axis which is called its axis or line of symmetry. The lowest point on the parabola (which in this case is the origin) is called its vertex.

We'll use this graph and equation to illustrate some ideas that have much wider applicability. Let's consider making some changes:

Rescaling

Suppose we multiply the value of $ y $ with a constant. Let's call it $ a $. Thus we obtain the new equation

$\displaystyle y= ax^2.\qquad(*) $

If $ a=1 $ this is the same equation as before. Let's consider some other possibilities, however:


Figure 2. Several parabolas

Figure 2 shows the graphs of these equations for

$\displaystyle a=-8,-4,~-2,~-1,~\-\frac{1}{2},~-\frac{1}{4},~\frac{1}{4},~\frac{1}{2}, ~1,~2,~4,~8. $

Graphs can be similarly reflected, stretched or compressed in the horizontal direction by multiplying $ x $ with a constant. However, in the present simple example this effect is equivalent to a vertical rescaling since $ (bx)^2 = b^2x^2 =ax^2 $ where $ a=b^2 $.

Translations

Consider now the effect of subtracting a constant $ k $ from $ y $:

$\displaystyle y-k = x^2. $

This is equivalent to

$\displaystyle y= x^2+k $

and so it has the effect of raising the graph of $ y=x^2 $ by $ k $ units. Of course, if $ k $ is negative, the graph is lowered (by $ \vert k\vert $ units).

Subtracting a constant $ h $ from $ x $ has the same effect in the horizontal direction. Consider the equation

$\displaystyle y = (x-h)^2\qquad(**) $

and compare it with the original equation $ (*) $. As in $ (*) $ , $ y $ is never negative. It assumes its minimum value ($ 0 $) at $ x=h $. Thus the graph of $ (**) $ is that of $ (*) $ shifted $ h $ units to the right. If $ h $ is negative, the graph is shifted $ \vert h\vert $ units to the left.


Figure 3. Translated Parabolas

Horizontal and vertical shifts can be combined, and any such combination is called a translation. The general form of a translation applied to $ (*) $ is given by

$\displaystyle y-k = (x-h)^2 $

which is often written as

$\displaystyle y= (x-h)^2 + k. $

A translation does not change the shape or orientation of a graph. It only changes its location. Figure 3 shows translations of the graph of $ (*) $ with

$\displaystyle (h,k) = (-1.5,-3),~(-1,-2),~(-0.5,-1),~(0,0),~(0.5,1),~(1,2),~(1.5,3). $

The Completed Square Form

The completed square form of a quadratic function $ f $ is

$\displaystyle f(x) = a(x-h)^2 + k.\qquad(***) $

Its graph is a parabola whose vertex is $ (h,k) $. If $ a>0 $ it opens up, if $ a<0 $ it opens down. (If $ a=0 $ the graph is a horizontal line which you can think of a s a degenerate parabola.) The vertical line $ x=h $ is the axis or line of symmetry of the parabola.

It is clear how a general quadratic polynomial

$\displaystyle f(x) = ax^2 + bx + c $

can be converted to the standard form. Just complete the square, as illustrated in the following example:

$\displaystyle \begin{array}{rclcl} y &=& -2x^2-4x+6 &\vert&\hbox{factor out \( ... ...uare} \\ &=& -2(x+1)^2 +8 &\vert& \hbox{perfect square form} \\ \end{array}$

The graph of this particular equation is given in Figure 4. The resulting parabola has $ a=-2 $ and the vertex ($ -1,8) $. It is obtained from the standard parabola $ (*) $ by reflecting it thorough the $ x $-axis, stretching it vertically and translating it 1 unit to the left and 8 units up. Note that all of these properties can be immediately read off the equation

$\displaystyle y= -2(x+1)^2 +8 . $


Figure 4. Another Parabola

Scaling and Translating Graphs

The principles illustrated here apply to any equation, so let's restate them:

As usual, I recommend that you do not memorize these facts. Rather think about what's happening in terms of the interplay between the graph and the equation, and figure out the effect of what you are doing, or what you need to do, rather than relying on remembering arcane facts.