Content

Sketching polynomial functions

Although polynomial graphs come in many shapes and sizes, they can be sketched once we find a few of their features.

Although it's unnecessary in the previous example, we could use a sign diagram to investigate stationary points. We choose values of \(x\) before and after each stationary point, and consider whether \(f'(x) = 6(x+3)(x-5)\) is positive or negative, as shown below.

Sign diagram

value of \(x\)		−3		5
sign of \(f'(x)\)	+	0	−	0	+
slope of graph \(y=f(x)\)	\(\diagup\)	—	\(\diagdown\)	—	\(\diagup\)

From the sign diagram we see directly that \(x=-3\) is a local maximum and \(x=5\) is a local minimum. However in the example this is deduced from other information.

In summary, the following information may be useful when sketching the graph of a polynomial \(y=f(x)\):

• \(x\)-intercepts, obtained by solving \(f(x)=0\)
• \(y\)-intercept, obtained by evaluating \(f(0)\)
• behaviour as \(x \to \pm \infty\), obtained by considering the leading term of \(f(x)\)
• stationary points, obtained by solving \(f'(x)=0\)
• sign diagram for \(f'(x)\), obtained by substituting values into \(f'(x)\).

It's not always necessary to calculate all of these; often, as in the previous example, a graph can be sketched with less information.

Repeated roots

When a polynomial has a factor \((x-a)\) to a power greater than 1, we say \(a\) is a repeated root. If \((x-a)^2\) is a factor of \(f(x)\), we say \(a\) is a double root or a root of multiplicity 2. In general, if \((x-a)^m\) is a factor of \(f(x)\), we say \(a\) is a root of multiplicity \(m\). A root of multiplicity 1 is often called a simple root.

Consider the polynomial \(f(x) = (x-a)^m\), where \(a\) is a real number and \(m\) is a positive integer. Obviously \(f(x)\) has a root at \(x=a\) of multiplicity \(m\). The following three graphs show \(y=f(x)\) when \(m=1,2,3\).

Detailed description of diagrams

An important fact to note is that the sign of \(f(x)\) changes at \(x=a\) when \(m\) is odd, and does not change when \(m\) is even. (To see this, note that if \(x>a\), then \(x-a\) is positive and \((x-a)^m\) is also positive. If \(x<a\), then \(x-a\) is negative; if \(m\) is even, then \((x-a)^m\) is positive, while if \(m\) is odd, then \((x-a)^m\) is negative.)

This fact becomes relevant when we consider more general polynomials with repeated roots. Consider the polynomials \(f(x) = (x+2)^m (x-1)\), where \(m\) is a positive integer. Then \(f(x)\) has a root at \(x=-2\) of multiplicity \(m\). The graphs for \(m=2,3,4\) are shown below.

Detailed description of diagrams

First consider the polynomial \(f(x) = (x+2)^2 (x-1)\), which has a double root at \(x=-2\). Then \(f(-2) = 0\). Since the power 2 is even, the factor \((x+2)^2\) does not change sign at \(x=-2\); if \(x\) is slightly more or less than \(-2\), then \((x+2)^2\) is positive. For \(x\) close to \(-2\), the factor \(x-1\) is negative, and so \(f(x)\) is negative for \(x\) slightly more or less than \(-2\). That is, \(f(x)\) does not change sign at \(x=-2\).

Next, consider \(f(x) = (x+2)^3 (x-1)\), which has a triple root at \(x=-2\). As 3 is odd, the factor \((x+2)^3\) does change sign at \(x=-2\). For \(x\) near \(-2\), the factor \(x-1\) is negative. So \(f(x)\) changes sign at \(x=-2\).

The above argument relies only on whether the multiplicity of the root \(x=-2\) is even or odd. Using a similar argument, one can prove the following theorem.

Screencast of interactive 5 ,  Interactive 5

Theorem

Let \(f(x)\) be a polynomial with a root \(a\) of multiplicity \(m\). Then \(f(x)\) changes sign at \(x=a\) if and only if \(m\) is odd.

If a polynomial \(f(x)\) has a root at \(x=a\) of even multiplicity \(m\), then \(f(x)\) does not change sign at \(x=a\), and yet \(f(a)=0\). Hence there must be a turning point and we obtain the next theorem.

Theorem

Let \(f(x)\) be a polynomial with a root \(a\) of even multiplicity. Then the graph of \(y=f(x)\) has a turning point at \(x=a\).

Additionally, we note from the above graphs that a root of multiplicity 2 or more always appears to be a stationary point. We can also prove this as a theorem.

Theorem

Let \(f(x)\) be a polynomial with a root \(a\) of multiplicity 2 or more. Then the graph of \({y=f(x)}\) has a stationary point at \(x=a\).

Proof

• Since \(a\) has multiplicity at least 2, we know that \((x-a)^2\) is a factor of \(f(x)\). (Maybe higher powers of \((x-a)\) divide into \(f(x)\), but \((x-a)^2\) certainly does.)
• We can then write \(f(x) = (x-a)^2 g(x)\) for some polynomial \(g(x)\). Differentiate \(f(x)\) using the chain and product rules to obtain

\[ f'(x) = 2(x-a) g(x) + (x-a)^2 g'(x) = (x-a) \big[ 2 g(x) + (x-a) g'(x) \big]. \]

• Thus \(f'(a) = 0\) and so \(x=a\) is a stationary point. \(\Box\)

Screencast of exercise 11

Transformation methods

We've seen from the module Functions II that, if we take the graph of \(y=f(x)\) and dilate or translate it in the \(x\) and \(y\) directions, or reflect in the axes, then the result is a graph of \(y=af(b(x-c))+d\), for some real numbers \(a,b,c,d\).

We can sometimes use this idea in reverse, as in the following example. It's easier than using the standard method.

Next page - Content - Polynomials from graphs