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Wize High School Grade 12 Pre-Calculus Textbook > Polynomial Functions

Polynomial Graphs & Characteristics

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Polynomial Graphs & Characteristics

A polynomial function of degree 'nn' can be expressed as:
f(x)=axn+bxn−1+ ... +cx+d\displaystyle \boxed{f(x) =ax^n+bx^{n-1}+~...~+cx+d} , where a,b,c,d∈Ra, b, c, d\in\mathbb{R}.
  • The degree of the polynomial is 'nn'.
  • The leading coefficient is 'aa'.
  • A function is increasing xxwhenever & yy increase and decrease simultaneously.
  • A function is decreasing whenever xx is increasing & yy is decreasing (or vice versa).

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Positive, Odd Polynomials

When a>0  &  n is odda>0~~\&~~n~\text{is odd}:


f(x)=xf(x)=x3f(x)=x5Degree135SymmetryPointPointPointMin # of111X-IntMax # of135X-IntEnd Behaviourx→∞, y→∞x→∞, y→∞x→∞, y→∞x→−∞, y→−∞x→−∞, y→−∞x→−∞, y→−∞Increasing(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)DecreasingNANANA\begin{array}{|c|c|c|c|}\hline\\ &f(x)=x&f(x)=x^3&f(x)=x^5\\\\\hline\\ \textbf{Degree}&1&3&5\\\\\hline\\ \textbf{Symmetry}&\text{Point}&\text{Point}&\text{Point}\\\\\hline\\ \textbf{Min \# of}&1&1&1\\\textbf{X-Int}&\\\\\hline\\ \textbf{Max \# of}&1&3&5\\\textbf{X-Int}&\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow\infin&x\rightarrow\infin,~y\rightarrow\infin&x\rightarrow\infin,~y\rightarrow\infin\\&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin\\\\\hline\\ \textbf{Increasing}&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)\\\\\hline\\ \textbf{Decreasing}&\text{NA}&\text{NA}&\text{NA}\\\\\hline \end{array}

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Negative, Odd Polynomials

When a<0  &  n is odda<0~~\&~~n~\text{is odd}:

f(x)=−xf(x)=−x3f(x)=−x5Degree135SymmetryPointPointPointMin # of111X-IntMax # of135X-IntEnd Behaviourx→∞, y→−∞x→∞, y→−∞x→∞, y→−∞x→−∞, y→∞x→−∞, y→∞x→−∞, y→∞IncreasingNANANADecreasing(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)\begin{array}{|c|c|c|c|}\hline\\ &f(x)=-x&f(x)=-x^3&f(x)=-x^5\\\\\hline\\ \textbf{Degree}&1&3&5\\\\\hline\\ \textbf{Symmetry}&\text{Point}&\text{Point}&\text{Point}\\\\\hline\\ \textbf{Min \# of}&1&1&1\\\textbf{X-Int}&\\\\\hline\\ \textbf{Max \# of}&1&3&5\\\textbf{X-Int}&\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow-\infin&x\rightarrow\infin,~y\rightarrow-\infin&x\rightarrow\infin,~y\rightarrow-\infin\\&x\rightarrow-\infin,~y\rightarrow\infin&x\rightarrow-\infin,~y\rightarrow\infin&x\rightarrow-\infin,~y\rightarrow\infin\\\\\hline\\ \textbf{Increasing}&\text{NA}&\text{NA}&\text{NA}\\\\\hline\\ \textbf{Decreasing}&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)\\\\\hline \end{array}

Wize Concept
For odd polynomial functions, the minimum number of real roots is 1 and the maximum number of real roots is 'nn'.

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Positive, Even Polynomials

When a>0  &  n is evena>0~~\&~~n~\text{is even}:
f(x)=x2f(x)=x4Degree24SymmetryLineLineMin # of02X-IntMax # of04X-IntEnd Behaviourx→∞, y→∞x→∞, y→∞x→−∞, y→−∞x→−∞, y→−∞Increasing(0,∞)(0,∞)Decreasing(−∞,0)(−∞,0)\begin{array}{|c|c|c|}\hline\\ &f(x)=x^2&f(x)=x^4\\\\\hline\\ \textbf{Degree}&2&4\\\\\hline\\ \textbf{Symmetry}&\text{Line}&\text{Line}\\\\\hline\\ \textbf{Min \# of}&0&2\\\textbf{X-Int}\\\\\hline\\ \textbf{Max \# of}&0&4\\\textbf{X-Int}\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow\infin&x\rightarrow\infin,~y\rightarrow\infin\\&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin\\\\\hline\\ \textbf{Increasing}&(0,\infin)&(0,\infin)\\\\\hline\\ \textbf{Decreasing}&(-\infin,0)&(-\infty,0)\\\\\hline \end{array}
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Negative, Even Polynomials

When a<0  &  n is evena<0~~\&~~n~\text{is even}:
f(x)=−x2f(x)=−x4Degree24SymmetryLineLineMin # of02X-IntMax # of04X-IntEnd Behaviourx→∞, y→−∞x→∞, y→−∞x→−∞, y→−∞x→−∞, y→−∞Increasing(−∞,0)(−∞,0)Decreasing(0,∞)(0,∞)\begin{array}{|c|c|c|}\hline\\ &f(x)=-x^2&f(x)=-x^4\\\\\hline\\ \textbf{Degree}&2&4\\\\\hline\\ \textbf{Symmetry}&\text{Line}&\text{Line}\\\\\hline\\ \textbf{Min \# of}&0&2\\\textbf{X-Int}\\\\\hline\\ \textbf{Max \# of}&0&4\\\textbf{X-Int}\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow-\infin&x\rightarrow\infin,~y\rightarrow-\infin\\&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin\\\\\hline\\ \textbf{Increasing}&(-\infin,0)&(-\infin,0)\\\\\hline\\ \textbf{Decreasing}&(0,\infin)&(0,\infin)\\\\\hline \end{array}

Wize Concept
For even polynomial functions, the minimum number of real roots is 0 and the maximum number of real roots is 'nn'.

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Example: Polynomial Graphs & Characteristics

Based on the following graph:


Find the following:
  • Number of xx-intercepts
  • End Behaviour
  • Odd or Even Degree
  • Positive or Negative Leading Coefficient
  • Degree of Polynomial
Number of xx-intercepts: 4

End Behaviour:
x→∞, y→∞x\rightarrow\infin,~y\rightarrow\infin
x→−∞, y→∞x\rightarrow-\infin,~y\rightarrow\infin

Odd or Even Degree: Even

Positive or Negative Leading Coefficient: Positive

Degree of Polynomial: 4

Practice: Polynomial Graphs & Characteristics

Based on the graph below, answer the following questions.