Functions (Graphs and Behaviour)

Hong Kong

Stage 4 - Stage 5

Lesson

Variation is a critically important concept in the study of any of the Sciences. Variation deals with the way two or more variables interact with each other.

To start our discussion, think about the equation $xy=36$`x``y`=36, with $x$`x` and $y$`y` as positive variables. Since $36$36 is a constant, any increase in $x$`x` will cause a decrease in $y$`y`. so that their product remains at $36$36.

You may note that for $x=18$`x`=18, $y=2$`y`=2 and for $x=2$`x`=2, $y=18$`y`=18, but the concept of variation is not just about finding the value of $y$`y` for a given value of $x$`x`. The concept of variation is a little more subtle. Variation is about understanding the *nature of the change* in $y$`y` as changes in $x$`x` are made.

The question we need to ask is "*how does $y$ y change when changes in $x$x are made?*"

The first type of variation is known as direct variation.

One variable $y$`y` (or some function of that variable), varies *directly* with another variable $x$`x` (or some function of it) if a change in the direction of $x$`x` causes a change *in the same direction* of $y$`y`.

What this means is that when $x$`x` gets bigger, so does $y$`y`, and when $x$`x` gets smaller, so does $x$`x`.

As a simple example, consider the relationship given by $y=kx$`y`=`k``x` for some *constant of variation* $k$`k`. Here $y$`y` varies directly with $x$`x`, so that doubling $x$`x` will double $y$`y`, halving $x$`x` will halve $y$`y`, and adding, say $5$5 to $x$`x` will add $5k$5`k` to $y$`y`, so that $y=k\left(x+5\right)=kx+5k$`y`=`k`(`x`+5)=`k``x`+5`k`. The variable $y$`y` will always move in the same direction as $x$`x`.

Another example is the relation given by $y=3x^2$`y`=3`x`2 , for $x\ge0$`x`≥0. We can say that $y$`y` varies directly with the *square of* $x$`x`, so that when $x^2$`x`2 increases (or decreases) then so does $y$`y`.

When one variable varies directly with another, the graph will be a straight line with the gradient of the line referred to as the *constant of variation*.

In this last example, we could think of the graph of $y=3x^2$`y`=3`x`2 for $x\ge0$`x`≥0 as one half of a parabola with the vertex at $\left(0,0\right)$(0,0). However, if we wish to show the direct variation of $y$`y` with $x^2$`x`2, an alternative approach is to graph $y$`y` against $x^2$`x`2. The "$x$`x`" axis becomes the "$x^2$`x`2" axis, and the graph becomes a straight line with gradient $3$3.

The first example we introduced where $y$`y` varies according to $xy=36$`x``y`=36 was an example of inverse variation. In general terms, $y$`y`, or some function of $y$`y`, varies inversely with $x$`x`, or some function of $x$`x`, if a change in $x$`x` causes an inverse change in $y$`y`.

To explain this, in our example we could express $y$`y` explicitly, so that $y=36\left(\frac{1}{x}\right)$`y`=36(1`x`). This form makes it clear that $y$`y` varies *directly* with the quantity $\frac{1}{x}$1`x`.

Inverse Variation

If $y$`y` varies *directly *with $\frac{1}{x}$1`x`, then it is said to vary *inversely *with $x$`x`.

As $x$`x` increases, $y$`y` decreases but not uniformly - rather its rate of decrease depends on the size of $x$`x`. Again referring to our example, if $x$`x` increases from $2$2 to $3$3, then $y$`y` changes decreases rapidly from $\frac{36}{2}=18$362=18 to $\frac{36}{3}=12$363=12. However if $x$`x` increases from $20$20 to $21$21, then $y$`y` decreases at a much slower rate from $\frac{36}{20}=1.9$3620=1.9 to $\frac{36}{21}=1.714$3621=1.714.

Perhaps one of the most familiar instances of inverse variation can be seen with Ohm's Law. Ohm's Law states that the current $I$`I`* amperes* flowing through a conductor between two points is *directly proportional* to the potential difference (known as the voltage $V$`V` *volts*). The constant of variation is given by $R$`R`* ohms* the resistance across the circuit. We can think about resistance as the "rocks" in a stream slowing the flow of the current.

Ohm's Law is usually written as $V=IR$`V`=`I``R` , however we could just as well write the law as $I=\frac{V}{R}$`I`=`V``R`.

Now if we were to set up a situation where the voltage, instead of the resistance, was held constant, and allow the resistance to become variable, we would conclude that the current varies inversely with the resistance.

Increasing the resistance will decrease the current and vice-versa. Note that the increase in current is not uniform. At low resistances, any decrease in ohms causes large increases in current flow.

The typical graph of a quantity $y$`y` varying inversely with $x$`x` is hyperbolic. For most physical relationships, the graph is restricted to the first quadrant. For example it makes no sense to talk about negative resistances.

Suppose we knew that, for a certain circuit of fixed voltage, the current is measured as $10$10 amperes when a total of $24$24 ohms of resistance is in place.

Then, from $I=\frac{V_{circuit}}{R}$`I`=`V``c``i``r``c``u``i``t``R`, we have that $10=\frac{V_{circuit}}{24}$10=`V``c``i``r``c``u``i``t`24, and this means that the constant voltage can be determined as $240$240 volts. Thus, for this circuit, we have $I=\frac{240}{R}$`I`=240`R` and we can graph the current against the varying resistance.

Note how, as the resistance becomes close to zero, the current increases without bound. Notice that no matter how much resistance we apply to the circuit, there will always be some current flowing.

If $a$`a` varies directly with $x$`x` cubed, and $a=12$`a`=12 when $x=4$`x`=4:

find the constant of variation, $k$

`k`define $a$

`a`in terms of $x$`x`find the value of $a$

`a`when $x=2$`x`=2

If $a$`a` is inversely proportional to $x$`x`, and $a=20$`a`=20 when $x=10$`x`=10:

Find the constant of variation, $k$

`k`.Express $a$

`a`in terms of $x$`x`.Find the value of $a$

`a`when $x=5$`x`=5.

The mass in grams, $M$`M`, of a cube of cork varies directly with the cube of the side length in centimetres, $x$`x`. If a cubic centimetre of cork has a mass of $0.29$0.29:

find the constant of variation, $k$

`k`express $M$

`M`in terms of $x$`x`find the mass of a cube of cork with a side length of $8$8 centimetres correct to two decimal places if necessary.

The number of eggs, $n$`n`, used in a recipe for a particular cake varies with the square of the diameter of the tin, $d$`d`, for tins with constant depth.

If $2$2 eggs are used in a recipe for a tin with a diameter of $17$17 cm:

Find the exact value of the constant of variation, $k$

`k`.How many eggs, $n$

`n`, would be used for a tin with a diameter of $39$39 cm?Round your answer to the nearest egg.