# Maths Power Laws: X and Higher Polynomials

## Linear, quadratic and cubic powers

You already know about linear powers, such as \(\normalsize{y=x}\), quadratic powers such as \(\normalsize{y=x^2}\) and now also cubic powers such as \(\normalsize{y=x^3}\) . Here are their graphs: It is important to understand how higher powers of \(\normalsize{x}\) behave. The general pattern follows the three functions above, but generally as the exponent increases, the function becomes more extreme — and it gets very big more quickly as \(\normalsize{x}\) increases.## Higher powers of \(\normalsize{x}\)

Recall that \(\normalsize{x^0=1}\) for any \(\normalsize{x}\). So the function \(\normalsize{y=x^0}\) is the same as the function \(\normalsize{y=1}\) — that is, it is a constant function that does not change at all. In the following diagram, you can see the functions \(\normalsize{y=x^n}\) for \(\normalsize{n=0,1,2,3,4,5}\) and how they grow with \(\normalsize{x}\).Q1(E): In the above graph, which function is which?

*increasing*function. However if \(\normalsize{n}\) is even, then as \(\normalsize{x}\) increases, two things can happen: the function decreases while \(\normalsize{x}\) is negative, and then increases for positive \(\normalsize{x}\).

## Index laws

*index laws*

## Growth rates of powers

*growth rate*than \(\normalsize{y=x^2}\). This becomes ever more noticeable as the exponent \(\normalsize{n}\) in \(\normalsize{y=x^n}\) increases.

Q2(E): Which has a faster growth rate: the function \(7x^5\) or \(5x^7\)?Q3(M): What about when \(\normalsize x\) is getting closer to zero? Which of the functions \(7x^5\) and \(5x^7\) will decrease faster?

## Combining powers of \(x\)

## Polynomials as products of factors

## Shapes of polynomial functions

*the graph of a degree n polynomial can meet an arbitrary line in at most n points*. This is an important theorem. Another variant is that a polynomial of degree \(\normalsize n\) can change direction — from up to down, or from down to up, as we view it from left to right —

*at most \(\normalsize (n-1)\) times*.

## Answers

A1.The function \(\normalsize y=x^0=1\) is light blue (the horizontal line), \(\normalsize y=x^1=x\) is red (the straight line going through the point \(\normalsize{[0.5, 0.5]}\)), \(\normalsize y=x^2\) is yellow (one of the 2 ‘even’ functions — with all positive values), \(\normalsize y=x^3\) is orange (one of the 3 ‘odd’ functions — with negative values for x < 0), \(\normalsize y=x^4\) is green (again an even function) and \(\normalsize y=x^5\) is dark blue (again an odd function).A2.Since \(\normalsize 7\) is greater than \(\normalsize 5\) the function \(\normalsize x^7\) grows faster than \(\normalsize x^5\) as \(\normalsize x\) gets larger. The coefficients are relatively unimportant, so that also \(\normalsize 5x^7\) grows faster than \(\normalsize 7x^5\).A3.Since \(\normalsize 7\) is greater than \(\normalsize 5\) the function \(\normalsize x^7\) decreases faster than \(\normalsize x^5\) as \(\normalsize x\) gets smaller. So also \(\normalsize 5x^7\) decreases faster than \(\normalsize 7x^5\) as \(\normalsize x\) gets smaller.

#### Maths for Humans: Inverse Relations and Power Laws

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