Revision:OCR Core 1 - Transforming graphs

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Revision:OCR Core 1 - Transforming graphs

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > OCR Core 1 - Transforming graphs

1 10. Transforming graphs
2 Also See
3 Comments

10. Transforming graphs

This is often a topic that people have trouble with, and it is a rather useful one.

Translating graphs

A translation (in the Cartesian plane) is (in the terms that one might expect of a person whose knowledge of Mathematics is low) a "move". This implies that there is no rotation, reflection, or other operation, other than a "move" in either the x-direction, y-direction, or both.

One can represent these movements using a column-vector (a concept that should be revised by those readers who do not know the meaning, and do not believe that they will be able to deduce the meaning from the context of the use of these column-vectors later in these notes).

One can begin with the concept of a movement in the y-direction.

Consider a function of "x":

f(x)

Now, one wishes to graph this function, thus:

y = f(x)

Hence, one can say that for the values of "x", there is an output value, which is assigned to a variable "y", and then placed on the Cartesian plane in such a manner as to have some formal meaning.

Now consider what one expects to happen to the following:

y = f(x) + k .

The answer is relatively easy to see. One is getting the same output from the input of "x", however, before assigning it to "y", one is adding a constant value to it. Hence, in comparison to the previous graph ( y = f(x) ), this one will have every "y" value at a level of "k" higher than the original, leading to the rule:

A graph of y = f(x) + k , is a translation of $\left( \begin{array}{ccc} 0 \\ k \end{array} \right)$ with reference to the graph of .

One can see that this is true, and it is a helpful property to know in conjunction with the completion of the square technique (as one is able to obtain the y-coordinate for the minimum, or maximum value of the curve).

One might also wish to consider the effects of the following:

y = f(x + k) .

This is a slightly more complex problem, but perfectly understandable.

Consider that the function is being applied to "x", and the value is given to "y" in the original graph. In this translated graph it is different. The function is being applied to x + k , and the result is given as "y". This means that one can consider the following:

y = f(x - k)

On the original graph, this is a point that is "k" units in the x-direction of "x", however when the translation is applied, it becomes:

y = f(x - k + k) = f(x)

Suggesting that in the new graph, the same value of "x" will have the value of "y" which corresponded with the x - k value of "x"; hence one can conclude the following rule:

A graph of y = f(x + k) , is a translation of $\left( \begin{array}{ccc} -k \\ 0 \end{array} \right)$ with reference to the graph of y = f(x) .

Stretching graphs

Now that one has considered the effects of the addition to the "x" value, and also the the value of the function applied to "x", it may be of interest as to what the effects of multiplication in either situation will cause.

Consider the graph of:

y = f(x)

The first change one might consider is:

y = kf(x)

As with the situations for the addition to the output of the function, this is a relatively simple idea. One is taking the output of the same value of "x" and then multiplying it, prior to allowing "y" to be the result. This will cause some form of stretch.

Evidently, this stretch will be in the direction of the y-axis, and if one is to think about this situation, it will have a scale factor of the constant of multiplication; hence one can derive the rule:

A graph of y = kf(x) , is a stretch in the y-direction, scale factor "k", with reference to the graph of y = f(x) .

Another case to consider is the following:

y = f(kx)

In this case, one is taking the value of "x", and then multiplying by "k", before applying the function. Now consider the following point on the original graph:

$y = f(\frac{x}{k})$ ,

and the corresponding point on the changed graph:

$y = f(\frac{kx}{k}) = f(x)$

This shows that the points for which the original graph, and the changed graph coincide (in the values of "y") are those values of "x" for which the original graph has a value of "x", which is $\frac{1}{k}$ times the "x" value in the new graph.

This suggests that the following rule can be deduced:

A graph of y = f(kx) , is a stretch in the x-direction, scale factor $\frac{1}{k}$ , with reference to the graph of y = f(x) .

Reflecting graphs

The reflection of graphs is a simple concept.

Consider that the y-axis is basically a set of positive values, and negative values, separated (by the x-axis) at zero.

Now one can consider the effect of changing the whole value of "y" from positive, to the corresponding (magnitudinally equivalent) negative value.

So, to formalise this idea:

y = f(x)

y = -f(x)

It is evident that for any value of "y" (such that "y" is not zero) will change to the corresponding (in terms of magnitude) negative value. This suggests that the following rule can be deduced:

A graph of y = -f(x) is a reflection in the x-axis, of the graph of y = f(x) .

This is easily deduced, as the y-axis positive values are symmetrical to their negative counter-parts (in the context of magnitude) with the x-axis as a line of symmetry.

Now consider the change:

y = f(-x)

This is another easy situation. One should not become concerned with specific cases of y = f(x) , and merely consider the general case.

The x-axis is the same as the y-axis in terms of the symmetry that was discussed, however, it is such that the y-axis is the line of symmetry between the positive and negative values (with reference to the correspondence being magnitudinal). This would suggest that the following rule can be deduced:

A graph of y = f(-x) is a reflection in the y-axis, of the graph of y = f(x) .

One can combine these rules, but one must ensure that one does the operations in the correct order (as there are different results for different orders), however this is unlikely to be a Core 1 topic (as it is not in the Core 1 text-book, rather the Core 3 text-book).