6A - Square-root function

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.

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!

6A - VIDEO EXAMPLE 1:

Graph the following square root function:

6A - VIDEO EXAMPLE 2:

Graph the following square root function:

6A - VIDEO EXAMPLE 3:

Graph the following square root function:

6A - VIDEO EXAMPLE 4:

Graph the following square root function: