Turning Points & End Behavior

Turning Points

The graphs exhibiting functions of odd degree and even degree show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing.

A polynomial function of degree n has at most n − 1 turning points, with at least one turning point between each pair of successive zeros.

End Behavior

The end behavior of a polynomial graph is determined by the dominating term—that is, the term of greatest degree. A polynomial of the form LaTeX: f\left(x\right)=a_0x^n+a_{n-1}x^{n-1}+...+a_0f(x)=a0xn+an−1xn−1+...+a0

has the same end behavior as LaTeX: f\left(x\right)=a_nx^nf(x)=anxn.

Example

For example, LaTeX: f\left(x\right)=2x^3+8x^2+2x-12f(x)=2x3+8x2+2x−12 has the same end behavior as LaTeX: f\left(x\right)=2x^3f(x)=2x3.

It is large and positive for large positive values of x and large and negative for negative values of x with large absolute value. That is, it rises to the right and falls to the left.

The figure below shows that as x increases without bound, y does also. As LaTeX: x\longrightarrow\infty,\:y\longrightarrow\infty,\:x⟶∞,y⟶∞,and as LaTeX: x\longrightarrow-\infty,\:y\longrightarrow-\inftyx⟶−∞,y⟶−∞.

Graph showing end behavior.PNG

Example

LaTeX: f\left(x\right)=-x^3+2x^2-x+2\:f(x)=−x3+2x2−x+2has the same end behavior as LaTeX: f\left(x\right)=-x^3f(x)=−x3. As LaTeX: x\longrightarrow\infty,\:y\longrightarrow-\infty\:x⟶∞,y⟶−∞and as LaTeX: x\longrightarrow-\infty,\:y\longrightarrow\inftyx⟶−∞,y⟶∞.

A second graph showing end behavior.PNG

End Behavior of Graphs of Polynomial Functions

Suppose that LaTeX: ax^n\:axnis the dominating term of a polynomial function f of odd degree.

  1. If LaTeX: a>0,\:a>0,then as LaTeX: x\longrightarrow\infty,\:f\left(x\right)\longrightarrow\infty,\:x⟶∞,f(x)⟶∞,and as LaTeX: x\longrightarrow-\infty,\:f\left(x\right)\longrightarrow-\infty.x⟶−∞,f(x)⟶−∞.

    Therefore, the end behavior of the graph is of the type shown here.
    end behavior with a positive and n odd.PNG
    We symbolize it as symbol for end behavior when a is positive and n odd.PNG

  2. If LaTeX: a<0,\:a<0,then as LaTeX: x\longrightarrow\infty,\:f\left(x\right)\longrightarrow-\infty,\:x⟶∞,f(x)⟶−∞,and as LaTeX: x\longrightarrow-\infty,\:f\left(x\right)\longrightarrow\infty.x⟶−∞,f(x)⟶∞.
    Therefore, the end behavior of the graph is of the type shown here.
    end behavior when a is negative and n is odd.PNG

    We symbolize is as Symbol for end behavior when a is negative and n is odd.PNG

Suppose that LaTeX: ax^n\:axnis the dominating term of a polynomial function f of even degree.

  1. If LaTeX: a>0a>0, then as LaTeX: \mid x\mid\longrightarrow\infty,\:f\left(x\right)\longrightarrow\infty∣x∣⟶∞,f(x)⟶∞.

    Therefore, the end behavior of the graph is of the type shown here.
    end behavior when a is positive and n is even.PNG
    We symbolize it as symbol for end behavior when a is positive and n is even.PNG

  2. If LaTeX: a<0a<0, then as LaTeX: \mid x\mid\longrightarrow\infty,\:f\left(x\right)\longrightarrow-\infty∣x∣⟶∞,f(x)⟶−∞.
    Therefore, the end behavior of the graph is of the type shown here.
    end behavior when a is negative and n is even.PNG
    We symbolize it as symbol for end behavior when a is negative and n is even.PNG

Example

Consider the graphs labelled A through D below and the functions whose equations are listed. Match each equation to one of the graphs.

Graphs of four polynomial functions labelled A through D.PNG

  1. LaTeX: f\left(x\right)=x^4-x+5x-4f(x)=x4−x+5x−4
  2. LaTeX: g\left(x\right)=-x^6+x^2-3x-4g(x)=−x6+x2−3x−4
  3. LaTeX: h\left(x\right)=3x^3-x^2+2x-4h(x)=3x3−x2+2x−4
  4. LaTeX: k\left(x\right)=-x^7+x-4k(x)=−x7+x−4

Solutions

  1. LaTeX: f\left(x\right)=x^4-x+5x-4f(x)=x4−x+5x−4
    Because f is of even degree with positive leading coefficient, its graph is C.
  2. LaTeX: g\left(x\right)=-x^6+x^2-3x-4g(x)=−x6+x2−3x−4
    Because g is of even degree with negative leading coefficient, its graph is in A.
  3. LaTeX: h\left(x\right)=3x^3-x^2+2x-4h(x)=3x3−x2+2x−4
    Because function h has odd degree and has a dominating term positive coefficient, its graph is in B.
  4. LaTeX: k\left(x\right)=-x^7+x-4k(x)=−x7+x−4
    Because function k has odd degree and a dominating term with negative coefficient, its graph is in D.