5.3 Graphs of Polynomial Functions

Polynomial Function Graphs

Polynomial functions produce smooth, continuous curves whose shapes are determined by a few key properties: degree, leading coefficient, zeros, and multiplicity. Once you can read these properties from an equation, you can sketch a reasonable graph without plotting dozens of points.

Key Features of Polynomial Graphs

Zeros (x-intercepts) are the x-values where $f(x) = 0$. These are the points where the graph crosses or touches the x-axis. A polynomial of degree $n$ can have at most $n$ real zeros.

End behavior describes what happens to $f(x)$ as $x \to +\infty$ and $x \to -\infty$. It's controlled entirely by the leading term (the term with the highest power):

• Even degree, positive leading coefficient: Both ends rise (up on the left, up on the right)
• Even degree, negative leading coefficient: Both ends fall
• Odd degree, positive leading coefficient: Falls on the left, rises on the right
• Odd degree, negative leading coefficient: Rises on the left, falls on the right

Turning points are local maxima and minima where the graph changes direction. A polynomial of degree $n$ has at most $n - 1$ turning points. For example, a cubic ($n = 3$) can have at most 2 turning points, and a quartic ($n = 4$) can have at most 3.

Factoring for Polynomial Zeros

To find zeros, set $f(x) = 0$ and factor. For example, given $f(x) = 2x^3 - 8x$:

1. Factor out the GCF: $2x(x^2 - 4) = 0$

2. Factor the difference of squares: $2x(x - 2)(x + 2) = 0$

3. Set each factor equal to zero: $x = 0$, $x = 2$, $x = -2$

Common factoring techniques you'll use:

• Greatest common factor (GCF)
• Grouping
• Trinomial factoring
• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
• Sum/difference of cubes

Polynomial Degree and Graph Characteristics

The degree is the highest power of the variable. It tells you a lot about the graph:

• Maximum number of real zeros: equal to the degree
• Maximum number of turning points: degree minus 1
• End behavior pattern: even degree vs. odd degree (see above)

One common mistake: even-degree polynomials are not always symmetric about the y-axis, and odd-degree polynomials are not always symmetric about the origin. That symmetry only holds for functions with exclusively even-powered terms (like $f(x) = x^4 - 3x^2$) or exclusively odd-powered terms (like $f(x) = x^3 - 5x$). A polynomial like $f(x) = x^4 + 2x^3$ is even degree but has no y-axis symmetry.

Graphing Polynomials Using Properties

Follow these steps to sketch a polynomial graph:

1. Identify the leading term and determine end behavior. For $f(x) = -3x^4 + 5x^2 - 1$, the leading term is $-3x^4$, so both ends point downward (even degree, negative coefficient).

2. Find the y-intercept by evaluating $f(0)$. Here, $f(0) = -1$.

3. Find the zeros by setting $f(x) = 0$ and factoring (or using other methods).

4. Determine the multiplicity of each zero (see next section) to know whether the graph crosses or touches the x-axis at that point.

5. Plot these key points and connect them with a smooth curve, making sure the graph matches the end behavior and respects the maximum number of turning points.

Zero Multiplicity in Polynomial Graphs

Multiplicity is how many times a factor is repeated. In $f(x) = (x - 1)^2(x + 3)$, the zero $x = 1$ has multiplicity 2 and the zero $x = -3$ has multiplicity 1.

Multiplicity controls the graph's behavior at each zero:

• Multiplicity 1 (odd): The graph crosses the x-axis cleanly, like a line passing through.
• Multiplicity 2 (even): The graph touches the x-axis and bounces back, like the vertex of a parabola.
• Multiplicity 3 (odd): The graph crosses the x-axis, but with a flattened "S" shape at the crossing point.

The general rule: odd multiplicity = crosses, even multiplicity = touches and turns around. Higher multiplicity means the graph flattens out more near that zero.

Intermediate Value Theorem Application

The Intermediate Value Theorem (IVT) says: if $f(x)$ is continuous on $[a, b]$ and $k$ is any value between $f(a)$ and $f(b)$, then there's at least one $c$ in $[a, b]$ where $f(c) = k$.

Since all polynomials are continuous everywhere, the IVT always applies. Its most common use is confirming that a zero exists in an interval.

For example, suppose $f(1) = -2$ and $f(3) = 5$. Because the function goes from negative to positive, the IVT guarantees at least one zero between $x = 1$ and $x = 3$. You can narrow down the location by testing the midpoint ($x = 2$) and repeating. This is called the bisection method.

Predicting Polynomial Behavior

Pulling it all together, here's how to predict a polynomial's overall shape from its equation:

• The leading term dictates end behavior and the general "frame" of the graph.
• Odd-degree polynomials always have at least one real zero (because their ends go in opposite directions, the graph must cross the x-axis). Even-degree polynomials may have no real zeros at all.
• The coefficients of other terms affect steepness, the exact location of turning points, and vertical shifts, but they don't change the end behavior.
• Factor the polynomial when possible to find zeros and their multiplicities, then use those to fill in the middle of the graph between the two ends.

Polynomial Functions and Their Properties

A few foundational definitions to keep straight:

• A polynomial function has the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where the exponents are non-negative integers and the coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers.
• The degree is the highest exponent ($n$).
• The leading coefficient is $a_n$, the coefficient of the highest-degree term.
• All polynomial functions are continuous (no breaks, holes, or jumps) and smooth (no sharp corners). This is what makes the graphing techniques above work reliably.