3.2 Polynomial behaviour and graphing

The leading term, zeros, and multiplicities of a polynomial control its large-scale shape. A careful sketch combines these features with intercepts and sign information.

3.2.1 End behaviour and local extrema

End behaviour. The behaviour of a function as \(x\to\infty\) and as \(x\to-\infty\).

For a polynomial \(P(x)=a_nx^n+\cdots\), the leading term \(a_nx^n\) determines end behaviour. Even degree gives matching end directions; odd degree gives opposite directions. A positive leading coefficient points upward as \(x\to\infty\), while a negative one points downward. A degree-\(n\) polynomial has at most \(n-1\) local extrema.

Worked Example: Describing cubic end behaviour

For \(P(x)=2x^3-5x^2+x+1\), the leading term is \(2x^3\). Therefore, \(P(x)\to\infty\) as \(x\to\infty\) and \(P(x)\to-\infty\) as \(x\to-\infty\).

Try It. Describe the end behaviour of \(-4x^5+2x\).
Check Your Work As \(x\to\infty\), \(P(x)\to-\infty\); as \(x\to-\infty\), \(P(x)\to\infty\).

Worked Example: Bounding the number of turns

A fourth-degree polynomial can have at most three local extrema. A graph showing four distinct turning points therefore cannot represent a fourth-degree polynomial.

Try It. What is the maximum possible number of local extrema for a degree-7 polynomial?
Check Your Work Six.

3.2.2 Zeros, multiplicity, and a systematic sketch

Multiplicity. The number of times a factor corresponding to a zero occurs in a polynomial.

At a zero of odd multiplicity, a polynomial changes sign and crosses the axis. At a zero of even multiplicity, it touches the axis and turns back. To sketch, factor completely, identify zeros and multiplicities, find the \(y\)-intercept, determine signs between zeros, establish end behaviour, and connect the features smoothly.

Worked Example: Sketching a factored cubic

For \(P(x)=5(x+2)(x-1)(x-3)\), the zeros \(-2\), 1, and 3 all have multiplicity 1, so the graph crosses at each. The \(y\)-intercept is 30. Positive cubic end behaviour runs from lower left to upper right.

Try It. Describe the intercept behaviour of \(x(x-4)^2\).
Check Your Work It crosses at \(x=0\) and touches without crossing at \(x=4\).

Worked Example: Interpreting repeated process thresholds

For \(R(q)=-q^2(2q+3)(q-1)\), the zero at \(q=0\) has multiplicity 2, so the graph touches there. It crosses at \(q=-3/2\) and \(q=1\). The negative leading term is degree 4, so both ends point downward.

Try It. Describe the zeros and end behaviour of \(2(x+1)^2(x-3)\).
Check Your Work It touches at \(x=-1\), crosses at \(x=3\), and has positive odd-degree behaviour from lower left to upper right.

Applied Problem: Analyzing a treatment-response polynomial. A dimensionless response is \(R(q)=-0.01q(q-20)^2(q-50)\). Identify the zeros and their multiplicities, state where the graph crosses or touches the axis, and describe its end behaviour.

Check Your Work The zeros are 0 and 50 with multiplicity 1, and 20 with multiplicity 2. The graph crosses at 0 and 50 and touches at 20. The degree is 4 and the leading coefficient is negative, so both ends approach \(-\infty\).

3.2.3 Practice Problems

  1. Describe the end behaviour of \(3x^4-2x+1\).
    Check Your Work Both ends approach \(\infty\).
  2. Describe the end behaviour of \(-x^3+4x\).
    Check Your Work Left end up, right end down.
  3. State the maximum number of local extrema for degree 6.
    Check Your Work Five.
  4. Identify zeros and multiplicities of \((x+2)^3(x-5)^2\).
    Check Your Work \(-2\) has multiplicity 3; 5 has multiplicity 2.
  5. Find the \(y\)-intercept of \(2(x-1)(x+3)\).
    Check Your Work \(-6\).
  6. A response model is \(P(q)=q(q-10)(q-25)\). At which flows does it cross zero?
    Check Your Work At \(q=0\), 10, and 25.
  7. A model contains \((q-18)^2\). Describe its graph at \(q=18\).
    Check Your Work The zero has even multiplicity, so the graph touches the axis without changing sign.
  8. Describe the end behaviour of the pump polynomial \(-0.2q^4+3q^2\).
    Check Your Work Both ends approach \(-\infty\).