In this explainer, we will learn how to evaluate the variation function at a point for a given function.

The variation is a number that measures the amount by which a function changes, when changes, within the domain of the function, from to , as shown in the diagram.

If we denote the change in by , we can write the variation as

The sign of also tells us whether the function increases , decreases , or stays the same when changes from to . More precisely, if we denote the change in by , the gradient of the line between the points and is

Since , the sign of the variation is the same as that of the gradient of the line between these two points.

As an example, consider the constant function , and letβs find the variation of this function when changes from 0 to 5:

As we would expect, since the function is a constant and therefore never changes, the variation between and is zero. This is true of the variation between any two values and :

Now, consider the linear function . Suppose we want to find the variation of this function when changes from 2 to 2.5. Since the slope of this line is , which is positive, we would expect the variation of the function to be positive between these values. We can simply substitute these values to find

This is the variation of the function when changes from 2 to 2.5. Since is positive, we know that the function increases between these two values of .

From , we can express in terms of and as

Thus, the variation can be written as

This is the variation of the function when changes by the amount starting from .

For the linear function when changes from 2 to 2.5 starting from , it changes by the amount

Using this, we can also express the variation of the linear function as which gives us the same result, as expected.

In general, since the amount by which changes, , is arbitrary, we can use a variable to express this with , and hence the variation , as a function of . This is known as the variation function of .

### Definition: The Variation Function

The variation function of a function at is defined as where represents the change in and represents the variation of the function from to .

In other words, the variation function measures the amount by which the function changes when changes from to , where the variable is the amount by which changes.

It is instructive to consider an example in which we identify the variation function algebraically when changes between two arbitrary values.

### Example 1: Identifying the Variation Function Algebraically

Fill in the blank: When changes from 2 to 2.3, the variation function for becomes .

### Answer

In this example, we want to find , the variation function, for an arbitrary function when changes from 2 to 2.3.

The variation function of a function at is defined as

If changes from 2 to 2.3, we have , the starting position, and , the amount by which changes from 2 to 2.3. On substituting these into , the variation function can be expressed algebraically in terms of 2 and 2.3 as

Letβs now consider the variation function of a linear function.

### Example 2: Finding the Variation Function of a Linear Function

Fill in the blank: If the function , then the variation function at .

### Answer

In this example, we want to determine the variation function of the linear function defined by at . We recall that the variation function of a function at is defined as

For at , the variation function is therefore

This tells us that for a linear function, the amount by which the function changes or the variation function is always the same no matter what the starting point is, and it is directly proportional to the amount by which the -values change, . This is expected as is a linear function, which defines a straight line.

In general, for the linear function , the variation function at is given by

The variation function of a linear function is proportional to the gradient of the line.

Now, letβs look at a few other examples in order to practice and deepen our understanding of variation functions. In the next example, we will find the variation function of a quadratic function.

### Example 3: Finding the Variation Function of a Quadratic Function

Determine the variation function for at .

### Answer

In this example, we will determine the variation of the quadratic function at .

We recall that the variation function of a function at is defined as

For , the variation function at is therefore

Now, letβs find the variation of a different quadratic function.

### Example 4: Finding the Variation Function of a Quadratic Function

Determine the variation function for at .

### Answer

In this example, we want to determine the variation of the quadratic function at . We recall that the variation function of a function at is defined as

For , the variation function at is therefore

In the next example, we will find the variation function of another quadratic function, but this time we will also evaluate it at a particular value of , the change in .

### Example 5: Finding the Value of the Variation Function of a Quadratic Function

If is the variation function for , what is when ?

### Answer

In this example, we will determine the variation function of the quadratic equation at and evaluate this function for a change in of . We recall that the variation function of a function at is defined as

The variation function of at is therefore

We can now evaluate this variation function at to find

This means that for a change in of from , the given function, , decreases by an amount of 2.36 units.

Now, letβs look at an example in which we find the variation function of a trigonometric function.

### Example 6: Finding the Variation Function of a Trigonometric Function

Determine the variation function of at .

### Answer

In this example, we want to find the variation function of a trigonometric function at . We recall that the variation function of a function at is defined as

For at , the variation function is therefore

Using and the cofunction identity,

The variation function becomes

In the next example, we will find the variation function of an exponential function.

### Example 7: Finding the Variation Function of an Exponential Function

Determine the variation function of at .

### Answer

In this example, we want to determine the variation function of the exponential function at . We recall that the variation function of a function at is defined as

For , the variation function at is therefore

Now, letβs find the variation function of a quadratic function and use this to find the value of an unknown coefficient in the quadratic.

### Example 8: Determining the Variation Function of a Quadratic Function and Finding the Value of One of Its Unknowns

Determine the variation function for at . Additionally, find if .

### Answer

In this example, we want to determine the variation function of at then use the given value to determine the unknown coefficient .

Recall that the variation function of a function at is defined as

For , the variation function at is therefore

Now, we can determine the coefficient by substituting into the variation function:

Thus, gives us the equation

In order to find the value of , we can multiply this by to give

On rearranging this to make the subject, we find

Therefore, to two decimal places, we have

Now, letβs look at an example in which we find the variation function of a trigonometric function and use this to find the value of an unknown coefficient.

### Example 9: Finding the Variation Function of a Trigonometric Function and Finding the Value of One of Its Unknowns

Determine the variation function of at .

If , find .

### Answer

In this example, we want to find the variation function of at then use the given value to determine the unknown coefficient . We recall that the variation function of a function at is defined as

For , the variation function at can be found to be

Using and the identity , the variation function becomes

If , we can substitute this value to find

Solving for then gives us

In the next example, we want to determine an unknown value using a given variation function at that value and comparing it with the variation function that we find directly from the given function.

### Example 10: Finding the Values of an Unknown given a Quadratic Function and Its Variation Function

If the variation function of at is , what is the value of ?

### Answer

In this example, we are given the variation function of a particular function at , and by comparing our result from the formula for a variation of a function with the stated variation function, we will determine the value of .

Recall that the variation function of a function at is defined as

For , the variation function at is therefore

Comparing this result with the given variation function, , we have

Since and is arbitrary, we must have

In the final example, we will find the variation function of a quadratic equation and use this with a given value of the function to determine the value of two unknowns that appear as coefficients of the function.

### Example 11: Finding the Variation Function of a Quadratic Function Then Determining the Values of Its Constants

Determine the variation function for at , and, from and , determine the constants and .

### Answer

In this example, we want to find the variation function of at and use and to determine the unknown constants and .

Recall that the variation function of a function at is defined as

For , the variation function at is therefore

We can use and to determine the constants and by forming two simultaneous equations. In particular,

Therefore, we have to solve the simultaneous equations

Rearranging the first equation gives us , and substituting this into the second equation gives us

Solving this for , we find

Substituting this value back into the first equation gives us

Thus, with the variation function , we find that

The variation function is also related to the average rate of change of a function defined by and the instantaneous rate of change, also known as the first derivative of at ,

However, these are beyond the scope of this explainer and will be covered elsewhere in more detail.

### Key Points

- The variation of a function is a number that measures the amount by which a function changes from to .
- The sign of the variation indicates in which overall direction a
function changes between the two points and
, and it is the same sign as the slope or
gradient of the line between these points. In particular, between and ,
- if , then is increasing,
- if , then is decreasing,
- if , then does not change.

- The variation function of a function at is defined as This is a measure of how much the function changes when changes from to or, in other words, when starts from and changes by a variable amount .
- A variation function can be used to determine an unknown coefficient or starting value for various functions when we are given the variation at a particular value for (i.e., ), or other information about the function .