Polynomial Graphs Notes

\begin{definition}[Polynomial Graphs] The graph of a polynomial function \$f(x)\$ is the set of all points \$(x, y)\$ in the coordinate plane such that \$y = f(x)\$. \end{definition}

\begin{callout}[type=note] The graph of a polynomial function is a smooth and continuous curve with no breaks, holes, or sharp corners. \end{callout}

Graphing Polynomial Functions

To graph a polynomial function:

1. Determine the end behavior of the graph.
2. Find the x-intercepts and y-intercept.
3. Identify any turning points or symmetry.
4. Plot additional points if necessary.
5. Sketch the graph.

End Behavior

The end behavior of a polynomial function describes how the graph behaves as \$x\$ approaches positive or negative infinity. It is determined by the degree and leading coefficient of the polynomial.

\begin{callout}[type=note] The end behavior of a polynomial function is determined by the degree and leading coefficient of the polynomial. \end{callout}

Even Degree

• If the leading coefficient is positive, the graph rises on both ends.
• If the leading coefficient is negative, the graph falls on both ends.

Odd Degree

• If the leading coefficient is positive, the graph falls on the left and rises on the right.
• If the leading coefficient is negative, the graph rises on the left and falls on the right.

Intercepts

X-Intercepts

The x-intercepts of a polynomial function are the real roots of the polynomial. They can be found by factoring the polynomial or using the Rational Root Theorem.

\begin{callout}[type=note] The x-intercepts of a polynomial function are the real roots of the polynomial. \end{callout}

Y-Intercept

The y-intercept of a polynomial function is the constant term of the polynomial. It can be found by evaluating the polynomial at \$x = 0\$.

\begin{callout}[type=note] The y-intercept of a polynomial function is the constant term of the polynomial. \end{callout}

Turning Points

The turning points of a polynomial function are the points where the graph changes direction. A polynomial of degree \$n\$ can have at most \$n - 1\$ turning points.

\begin{callout}[type=note] A polynomial of degree \$n\$ can have at most \$n - 1\$ turning points. \end{callout}

Symmetry

Some polynomial functions have symmetry:

• Even functions are symmetric about the y-axis.
• Odd functions are symmetric about the origin.

\begin{callout}[type=note] Some polynomial functions have symmetry:

• Even functions are symmetric about the y-axis.
• Odd functions are symmetric about the origin. \end{callout}

\begin{callout}[type=example] Graph the polynomial function \$f(x) = x^3 - 3x^2 + 2x\$.

1. End Behavior: The polynomial has an odd degree and a positive leading coefficient, so the graph falls on the left and rises on the right.
2. X-Intercepts: Factoring the polynomial gives \$f(x) = x(x - 1)(x - 2)\$, so the x-intercepts are \$(0, 0)\$, \$(1, 0)\$, and \$(2, 0)\$.
3. Y-Intercept: The y-intercept is \$f(0) = 0\$.
4. Turning Points: The polynomial has degree 3, so it can have at most 2 turning points.
5. Symmetry: The polynomial is odd, so it is symmetric about the origin.

The graph is shown below:

\begin{placeholder}[type=graph, description=Graph of the polynomial function f(x) = x^3 - 3x^2 + 2x, showing the x-intercepts, y-intercept, and end behavior.] \end{placeholder} \end{callout}

\begin{callout}[type=self_review]

1. Graph the polynomial function \$f(x) = x^4 - 4x^2\$.
2. Graph the polynomial function \$f(x) = -x^3 + 3x^2 - 2x\$.
3. Graph the polynomial function \$f(x) = x^5 - 5x^3 + 4x\$. \end{callout}