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Graph Translations and Stretching

When we are working with a graph of any kind, there are many ways we can transform it but preserve its family.  For example, we can translate a parabola (move it around on the coordinate system), or stretch it uniformly, and still have a parabola.  In this tutorial, we will examine translations and stretches.

Translation

Translating a graph means moving it vertically and/or horizontally around the coordinate plane.
 
For instance, say I have a parabola given by , and I want to shift it so the vertex is now at the point  instead of at the origin .  In order to do that, I take the function, and I change it as follows: 

which is the same, in this case, as

 

In general, then, in order to shift a given function  by  units horizontally and  units vertically, we replace  by , and add .  If  is a positive number, the translation is to the right.  If  is negative, then the translation is to the left.  If  is positive, the translation is upward, while if  is negative, the translation is downward.

Now, let us consider the translation of a circle.  The equation for a circle with radius r, centered at the origin, is .  In order to find the equation of a circle with the same radius, but now with center (h,k), we translate the original equation in this way:


           
This shifts the circle by h units along the x-axis and by k units on the y-axis, thus centering it at the point (h,k).  For example, a circle of radius 1, centered at the point (-3, 4) is given by:

, or .


In general, then, the graph of a relation is shifted by  units horizontally and by  units vertically if we replace  by  and  by .

Stretching

In addition to changing the “location” of the graph, we may also change the way it appears.  We can do this by either stretching it, or squashing it, in relation to the y-axis.  In order to stretch it, all we need do is take our function , and multiply by some constant to get .  This has a stretching effect, because it results in taking all of the points on the graph of y = g(x) and multiplying the y-coordinates by the same number a.

With a line, you may not notice quickly what has happened, but look at the difference in these parabolas

 

All three functions are of the form , just with differing values of . Notice that the points (1, 1) and
(-1, 1) are on the graph of f(x) = x2, the red graph above.  The purple graph contains the points (1, 3) and (-1, 3), the result of multiplying the y-coordinates on the graph of f(x) = x2 by 3.  Of course, the purple graph has equation y = 3x2.  It might be said to be “steeper” than the red graph (we say that it was stretched vertically), while the green graph,
y = .25x2, looks flatter or “squashed” vertically.

 

For values of , consider the difference in these parabolas

 

The dashed red line is , which is also the image of the graph of the function  reflected across the x-axis.  Notice that the shapes of all three “stretchings” of  in this image are the same as in the previous example, except that all have been reflected across the x-axis.

Example

Find the equation of a parabola with vertex  that goes through the point .

Solution

First, since this is a parabola, we know that our equation is going to be of the form .  Now, since our vertex was given as , we know what values  and  have.


So, our equation is now:

.

Now, we need to find  so that our parabola goes through the point .  In order to do that, we just plug it in and solve.

The equation for our parabola is .