Polynomial Graph

Graphing Polynomial Functions

The graph of a polynomial function provides a visual representation of how the function's value changes as the input value $x$ changes. The shape of this graph can vary greatly depending on the degree and coefficients of the function.

Point Plotting Method

The most fundamental way to draw a graph is by determining several $(x, y)$ point pairs that satisfy the function, then connecting them with a smooth curve.

Steps:

1. Choose several different values for $x$.
2. Calculate the value $y = P(x)$ for each chosen $x$ value.
3. Create a table of $(x, y)$ value pairs.
4. Plot these points on the coordinate plane.
5. Connect the points with a smooth and continuous curve.

Linear Function (Degree 1)

Graph the function $f(x) = 2x + 5$.

We choose several $x$ values and calculate $y$:

Plot the points and connect them:

Quadratic Function (Degree 2)

Graph the function $g(x) = x^2 - 2x - 3$.

Table of values:

Plot the points and connect with a smooth curve (parabola):

Cubic Function (Degree 3)

Graph the function $h(x) = x^3 + 3x^2 - 4$.

Table of values:

Plot the points and connect with a smooth curve:

General Characteristics of Polynomial Graphs

Graphs of polynomial functions are always smooth (no sharp corners) and continuous (no jumps or breaks). Their general shape is heavily influenced by the degree of the polynomial.

• Degree 0: $P(x) = c$. The graph is a horizontal line.
• Degree 1: $P(x) = ax + b$. The graph is a straight (slanted) line.
• Degree 2: $P(x) = ax^2 + bx + c$. The graph is a parabola.
• Degree 3: $P(x) = ax^3 + \dots$. The graph has a shape like the letter 'S' or a reverse 'S', and can have up to two 'peaks' or 'valleys'.
• Degree 4: The graph can have up to three 'peaks' or 'valleys'.
• Degree 5: The graph can have up to four 'peaks' or 'valleys'.

In general, the graph of a polynomial function of degree $n$ can intersect the $x$-axis at most $n$ times and has at most $n-1$ turning points (peaks or valleys).

End Behavior

One important characteristic of polynomial graphs is their end behavior, which describes the direction of the graph as $x$ approaches positive infinity ($x \to \infty$) or negative infinity ($x \to -\infty$).

The end behavior is determined solely by the leading term $a_n x^n$:

1. Degree $n$ (Even or Odd)
2. Sign of the Leading Coefficient $a_n$ (Positive or Negative)

There are four possible combinations:

1. $n$ Even, $a_n > 0$ (Positive):

   • As $x \to \infty$, $y \to \infty$ (rises right $\nearrow$)

   • As $x \to -\infty$, $y \to \infty$ (rises left $\nwarrow$)

   • Examples: $y = x^2$, $y = x^4$

     Graph of $y = x^2$ (Even, Positive)

     Graph rises to the left and right.

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2. $n$ Even, $a_n < 0$ (Negative):

   • As $x \to \infty$, $y \to -\infty$ (falls right $\searrow$)

   • As $x \to -\infty$, $y \to -\infty$ (falls left $\swarrow$)

   • Examples: $y = -x^2$, $y = -x^6$

     Graph of $y = -x^2$ (Even, Negative)

     Graph falls to the left and right.

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3. $n$ Odd, $a_n > 0$ (Positive):

   • As $x \to \infty$, $y \to \infty$ (rises right $\nearrow$)

   • As $x \to -\infty$, $y \to -\infty$ (falls left $\swarrow$)

   • Examples: $y = x$, $y = x^3$, $y = x^5$

     Graph of $y = x^3$ (Odd, Positive)

     Graph falls to the left and rises to the right.

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4. $n$ Odd, $a_n < 0$ (Negative):

   • As $x \to \infty$, $y \to -\infty$ (falls right $\searrow$)

   • As $x \to -\infty$, $y \to \infty$ (rises left $\nwarrow$)

   • Examples: $y = -x$, $y = -x^3$

     Graph of $y = -x^3$ (Odd, Negative)

     Graph rises to the left and falls to the right.

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Using End Behavior

Knowing the end behavior is very helpful for identifying the graph of a polynomial function without having to plot it in detail.

Application Example:

Match the following functions with their likely end behavior:

1. $f(x) = x^4 + 2x^3 - 2x - 3$
   • Leading term: $x^4$
   • Degree $n=4$ (Even)
   • Leading coefficient $a_n=1$ (Positive)
   • End behavior: Rises left ($\nwarrow$), Rises right ($\nearrow$)
   Graph of $f(x) = x^4 + 2x^3 - 2x - 3$

   End Behavior: $\nwarrow$ $\nearrow$

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2. $g(x) = -x^3 + 2x^2 - x + 1$
   • Leading term: $-x^3$
   • Degree $n=3$ (Odd)
   • Leading coefficient $a_n=-1$ (Negative)
   • End behavior: Rises left ($\nwarrow$), Falls right ($\searrow$)
   Graph of $g(x) = -x^3 + 2x^2 - x + 1$

   End Behavior: $\nwarrow$ $\searrow$

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3. $h(x) = -x^6 - \frac{11}{4}x^5 + x^4 + 5x^3 + 2$
   • Leading term: $-x^6$
   • Degree $n=6$ (Even)
   • Leading coefficient $a_n=-1$ (Negative)
   • End behavior: Falls left ($\swarrow$), Falls right ($\searrow$)
   Graph of $h(x) = -x^6 - \frac{11}{4}x^5 + x^4 + 5x^3 + 2$

   End Behavior: $\swarrow$ $\searrow$

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4. $k(x) = 25x^5 - 20x^4 - 26x^3 + 12x^2 + 9x - 1$
   • Leading term: $25x^5$
   • Degree $n=5$ (Odd)
   • Leading coefficient $a_n=25$ (Positive)
   • End behavior: Falls left ($\swarrow$), Rises right ($\nearrow$)
   Graph of $k(x) = 25x^5 - 20x^4 - 26x^3 + 12x^2 + 9x - 1$

   End Behavior: $\swarrow$ $\nearrow$

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By analyzing the leading term, we can predict the general shape of the graph at its ends.