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Revision:OCR Core 1 - Some important graphs

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

1 3. Some important graphs
2 Also See
3 Comments

3. Some important graphs

The idea of a function

A very simple way to think of a function is a machine. A when one inputs a value of a variable into the machine, it outputs a unique value. Consider the well known graph of $y = x^{2}$ , this is an example of a function. In the case of the aforementioned equation, the function of is $x^{2}$ , and the "output" is . One should not say "the function is $x^{2}$ " as the function is not this, the function is the action of squaring the input. One can denote a function in the following manner:

$f(x) = x^{2}$

Any letter can be used in the place of "f", however generally one will see "f" in situations where this does not conflict with other information (in other terms, a situation where there is no constant, variable, or other function denoted by "f", such that there is no ambiguity when one uses the letter).

One can use this notation to shorten writing as one need not write out the whole function over, and over again; for instance:

$f(x) = x^{3} + x^{2} + x + 1$

Now:

$f(3) = 3^{3} + 3^{2} + 3 + 1 = 27 + 9 + 3 + 1 = 40$ .

One can define functions using a flow chart.

Graphs of positive integer powers of "x"

The graph of $y = x^{2}$ is a well known one, the parabolic curve and symmetry about the y-axis is also present in other graphs.

Observe the following graphs.

Figure 003

One can tell that the graphs of y = x^2 , y = x^3 , and y = x^4 (shown in the above graph) all have similarities.

All of the graphs go through the point (1, 1). This is obviously true as if one raises 1 to the power of anything, it is 1.

One can also notice that the graphs of y = x^2 , and y = x^4 are symmetrical about the y-axis, and have similar shapes, however the graph of y = x^3 is different.

There is a trend of flattening that is appearing. Consider the following:

$Let\ f(x) = x^{2}$

$Let\ g(x) = x^{3}$

$Let\ h(x) = x^{4}$

Now consider what happens for 0 < x < 1 .

Consider the value, x = 0.5 :

f(0.5) = 0.25

g(0.5) = 0.125

h(0.5) = 0.0625

0.25 > 0.125 > 0.0625

However, it is known that all of the graphs go through the point (1, 1). To generalise this, one can say:

n > 1

$x = 1 - \frac{1}{n}$

Hence:

0 < x < 1

$f(x) = \left( 1 - \frac{1}{n} \right) ^{2} = 1 - \frac{2}{n} + \frac{1}{n^{2}}$

$g(x) = \left( 1 - \frac{1}{n} \right) ^{3} = 1 - \frac{3}{n} + \frac{3}{n^{2}} - \frac{1}{n^{3}}$

$h(x) = \left( 1 - \frac{1}{n} \right) ^{4} = 1 - \frac{4}{n} + \frac{6}{n^{2}} - \frac{4}{n^{3}} + \frac{1}{n^{4}}$

Now, define a function:

$k(x) = x^{m}$

As $m \to \infty$ the value of k(x) for 0 < x < 1 , the value of k(x) will become smaller.

It is known that all of these graphs go through (1, 1), and therefore this explains the changing shape. The higher the power of the function, the steeper the rise from a value 0 < x < 1 to the known value (1, 1). This causes a shape more and more like a right-angle to appear with the later powers.

Another observation that can be made in more detail is that the even powers of "x" cause a graph with symmetry about the y-axis, and the odd functions are not symmetrical about the y-axis.

There is a simple explanation:

$- \infty < x < 0$

$m \equiv 0 \pmod {2}$

$k(x) = x^{m}$

Hence:

$k(x) = x \times x \times x \times x \times ... \times x$

As x < 0 , and $m \equiv 0 \pmod {2}$ , it follows that k(x) > 0 .

Therefore, for $x \in \mathbb{R}$ , the graph y = k(x) is above the x-axis (touching at the Origin), and also the graph is symmetrical about the y-axis.

Consider the same values of "x", and the same function, but $m \equiv 1 \pmod {2}$ . This means that k(x) < 0 , and hence the graph of y = k(x) can be below the x-axis, and is not symmetrical about the y-axis.

Graphs of the form $y= ax^{2} + bx + c$

The graphs of the form $y = ax^{2} + bx + c$ are parabolic, and have lines of symmetry that are vertical. One can observe one example in the graph of $y = x^{2}$ whose line of symmetry is the y-axis (or x = 0 ).

Now one can consider the effects on the graph of changing the values of a, b, and c. Evidently changing c will cause the graph to move in the y-direction. Changing b will cause a move in the x-direction, and the sign of a dictates the orientation of the graph. A curve where a > 0 will curve "upwards", whereas a curve where a < 0 will curve "downwards".

The point of intersection of two graphs

This is much like finding the point of intersection of two lines. One can deduce that if there are two functions which when a value is input, will produce an identical output, these functions are equal. This information can then be used to conclude upon values for which the resultant equation (formed from equating the functions) is satisfied.

Example

1. Find the coordinates of the points at which the line y = x , and the curve $y = x^{2}$ intersect.

One simply has to equate the two:

$x^{2} = x$ ,

and then reduce this to a quadratic equation of the form $ax^{2} + bx + c = 0$ :

$x^{2} - x = 0$

Factorise:

x(x - 1) = 0

Therefore, x = 0, or x = 1.

Hence one can put these values into either of the original functions (both if one wishes to check one's values) to obtain the "y" values, and hence the coordinates.

Thus, the coordinates are: (0, 0), and (1, 1).

If one obtains a single point of intersection, it is proof that the line is a tangent to the curve at the point which one has calculated (this point is called the point of contact of the tangent to the curve).

Using factors to sketch graphs

To solve a quadratic equation, one can factorise it. By quadratic equation one refers (in this case) to an equation of the form $ax^{2} + bx + c = 0$ . This is important as it shows that when one obtains solutions through factorisation, one is obtaining the intercepts on the x-axis of the curve. Consider the previous section, one equates the two expressions of y in terms of x to obtain an equation whose solution is the intercepts of the two equations. In this example, one could think of the intercepts between $y = ax^{2} + bx + c$ , and y = 0 ; equating them gives: $ax^{2} + bx + c = 0$ .

It follows that if one can solve the equation, $ax^{2} + bx + c = 0$ , one knows of the x-axis intercepts (or lack thereof) for the function; these can be used (in conjunction with the sign of a, as mentioned previously, in order to give a sketch graph representation of the function $y = ax^{2} + bx + c$ ).