Validity, Formula, Graph

A function is a set of ordered pairs whereby for each element in the first place (domain) there is exactly one element in the second place (range).

When defining a function, it is necessary to consider what values are possible for the domain, in the context of the the purpose of the function.

In a real life context, we take data values, plot them and consider what kind of mathematical relation is similar to the data values. (The project). This is called modelling.

In a mathematical context, we are often able to calculate an algebraic relationship between the domain and the range.

Example: Large and Small Squares

Consider this square that measures 8 cm by 8 cm and has a smaller square of side length (p+1) cm cut out of it. How much area is remaining?

The area remaining depends on the value of p.

Let’s create a function R(p) that maps the value of p to the area remaining, that is, R.

Validity

First, we need to consider what values of p are possible in this context.

For a square to exist, we need that p+1 is bigger than (or equal to) zero. We can make sense of ‘no square’ but we can’t make sense, in the context of paper and scissors, of a square that has a negative side length.

We also require that p+1 \leq 8.

Therefore:

    \[0 \leq p+1 \leq 8\]

in other words,

    \[-1\leq p \leq 7\]

Ordered pairs

Second, let’s calculate some ordered pairs that are part of this function.

When p=-1, the side length of the small square is 0, now 0^2=0 so the area remaining is 64.

When p=0, the side length of the small square is 1, now 1^2=1 so the area remaining is 63.

When p=1, the side length of the small square is 2, now 2^2=4 so the area remaining is 60.

and so on….

We can plot the points \{(-1,64);(0,63);(1,60)\} on a graph.

Formula

Finally, since we are repeating the same calculation, let’s write it down in general:

We begin with area 64, and subtract the area of the small square.

The area of the small square = (p+1)^2=(p+1)(p+1)=p^2+2p+1

Therefore, our formula is:

    \begin{align*}R(p)&=64-(p^2+2p+1)\\[10pt]&=64-p^2-2p-1\\[10pt]&=63-2p-p^2\end{align*}

To write it accurately, we should say:

    \[R(p)=63-2p-p^2, \quad -1\leq p \leq 7\]

The graph (click on graph a few times):

To create this graph on GeoGebra, type in or copy/paste: R(p)=63-2p-p^2, -1<=p<=7 to the input bar. To plot the points, you may plot them individually, or you may plot them as a sequence: Sequence[(p,63-2p-p^2),p,-1,7]


Corner Cut Off

A rectangle is made of blue paper and measures 20 cm by 10 cm.

A corner is cut off.

The corner that is cut off is a right angled triangle, with shorter sides 2p cm and p-1 cm.

What values of p are valid?

Create a formula h(p) that maps the value p on the slider to the value Hypotenuse.

Create a formula r(p) that maps the value p on the slider to the value Area Remaining.

Check both formula using the value p=3.

Graph your function h(p) and your function r(p) on Geogebra for the valid values of p.

Solution:


Four Corners Cut Off

Four identical squares are cut from the corners of a piece of blue paper measuring 8 cm by 8 cm.

The side length of one the squares cut off is p cm.

What are the valid values of p?

Create a function A(p) that maps the value of p shown on the slider to the Area Remaining, remember to state the domain with the function.

Graph the function for values of p in the domain. State the range of the function.


Rectangle in Semi Circle

A rectangle is drawn inside a semi circle that has radius 5 cm.

 

Find a function that maps the value p on the slider to the area of the rectangle.

Graph the function for valid values of p. State the domain and the range of the function.


Inscribed Polygon

A regular polygon with p sides is drawn inside a circle with radius 5 as shown.

Show through calculation that the area of the polygon, when p = 7, is 68.41.

Calculate a function A(p) that maps the value p to the area of the polygon A.