New Zealand
Level 7 - NCEA Level 2

# Graphing Square Root Functions

Lesson

## Graphing tips

When graphing functions there are a number of useful non-calculus tools that you could consider. These could include:

• Recognising the basic form of the function
• Recognising embedded dilations and translations
• Considering the domain and range
• Plotting a few points
• Identifying any symmetrical features
• Finding the axes intercepts
• Looking at the behaviour of the function at its extremities

Relying solely on technology can be hazardous. Technology is a great tool, but it can only ever deliver a plot within a rectangular window that may or may not contain some or all of the graph's essential features.

Moreover, an important reason why we classify types of graphs is that they are useful for modelling physical phenomena - the path of a projectile, or a planet, the rise and fall of demand and supply, radioactive decay, depreciation and investment growth just to name a few.

It is thus vitally important to be able to recognise and select forms appropriate to that real world phenomena. This means developing a good understanding of each of the main forms and learning to utilise a few simple and universal tools to sketch them.

The above list is not exhaustive, and there is no implied order. In fact only some of the considerations might apply for any given function.

We will attempt to provide you with the language of mathematical investigation using two examples involving root functions.

## Example 1:

Consider the function given by $f\left(x\right)=-2\sqrt{3-x}$f(x)=23x .

The basic form of this function is the root function $y=\sqrt{x}$y=x, but clearly there are other complications to consider.

We need to ensure that the argument $3-x\ge0$3x0, and this means that the function's natural domain includes all real numbers $x$x where $x\le3$x3

We also note that when $x=3$x=3, $y=-2\times\sqrt{0}=0$y=2×0=0 and when $x=0$x=0$y=-2\sqrt{3-0}=-2\sqrt{3}$y=230=23. This means that the $x$x and $y$y intercepts are $\left(3,0\right)$(3,0) and $\left(0,-2\sqrt{3}\right)$(0,23).

As $x$x decreases, from $3$3 to $0$0 and into negative numbers, the quantity $3-x$3x increases, but the square root effect slows that increase down.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems