6A - Square-root function

Learning intentions:

In this section we will examine:

The square root graph and its transformations (dilations, reflections and translations).
How to graph the square root function from a given equation.

The square root graph

The square root function has a general form:

All square root graphs have the same shape, they are just transformed (dilated, reflected and translated) according to the values of the parameters a, n, h and k.

Introduction to the square root graph

Consider the following numbers and their square roots:

Table 1 - The square root of perfect squares

If we plot these point on a set of axes we can see the shape of the square root graph:

Figure 1 - The square root graph.

Reflection in the x-axis

A reflection in the x-axis occurs when occurs when there is a negative a term:

Figure 2 - The square root graph reflected in the x-axis.

Reflection in the y-axis

A reflection in the x-axis occurs when occurs when there is a negative n term:

Figure 3 - The square root graph reflected in the y-axis.

Looking ahead: Domain of a function

In the near future we will examine the domain of a function. The domain is the set of all of the x-coordinates (or first elements of an order pair). When we consider domains of functions they can be maximal/implied or they can be restricted.

For the square root function, the maximal domain is restricted to the expression under the square root being greater than or equal to zero. This is because you cannot have the square root of a negative number - it is undefined.

Transformations of the square root graph

When considering transformations of the square root graph, it is easiest to have the equation in the following form:

We can consider the effects of each parameter (a, n, h and k) on the square root graph.

a will cause a vertical dilation by a factor of a from the x-axis.
n will cause horizontal dilation - if n < 0 (negative) the graph is reflected in the y-axis.
h will cause a horizontal translation of h units.
k will cause a vertical translation of k units.

Examining the individual effects of a, h and k

The effects of the parameter a

The dynamic GeoGebra worksheet illustrates the effect of a on the square root graph

Please click on the play button in the bottom left hand corner to animate!

From the graph above we can see that:

When a > 1 the graph is steeper.
When 0 < a < 1 the graph is shallower.
when a < 0 (negative) the graph is reflected in x-axis.

a causes a dilation by a factor of a from the x-axis.

The effects of the parameter n

The dynamic GeoGebra worksheet illustrates the effect of n on the square root graph

Please click on the play button in the bottom left hand corner to animate!

From the graph above we can see that:

when n < 0 (negative) the graph is reflected in y-axis.

The effects of the parameter h

The dynamic GeoGebra worksheet illustrates the effect of h on the square root graph.

Please click on the play button in the bottom left hand corner to animate!

From the graph above we can see that:

When h > 0 (positive) the graph is translated h units in the negative x-direction.
When h < 0 (negative) the graph is translated h units in the positive x-direction.

The effects of the parameter k

The dynamic GeoGebra worksheet illustrates the effect of k on the square root graph.

Please click on the play button in the bottom left hand corner to animate!

From the graph above we can see that:

When k > 0 (positive) the graph is translated h units in the positive y-direction.
When k < 0 (negative) the graph is translated h units in the negative y-direction.

Examining the combined effects of a, h and k

The dynamic GeoGebra worksheet illustrates the combined effect of a, h and k on the square root graph.