6.1 Polynomial Functions and Their Graphs

Key Features of Polynomial Functions

A polynomial function is built from terms with non-negative integer exponents, combined using addition, subtraction, and multiplication. Examples range from simple linear functions like $f(x) = 3x + 1$ to more complex expressions like $f(x) = 2x^5 - x^3 + 4x - 7$. Two properties control most of what you need to know about a polynomial's graph: its degree and its leading coefficient.

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Degree

The degree of a polynomial is the highest exponent on the variable. It tells you:

• The maximum number of turning points the graph can have
• The maximum number of real zeros (x-intercepts)
• Whether the end behavior is "same on both sides" or "opposite on each side"

Leading Coefficient

The leading coefficient is the coefficient attached to the highest-degree term. It controls the direction of the end behavior. A positive leading coefficient means the graph ultimately rises on the right; a negative one means it ultimately falls on the right.

End Behavior of Polynomial Functions

End behavior describes what happens to $f(x)$ as $x \to +\infty$ and $x \to -\infty$. You only need the degree and leading coefficient to figure this out.

Even-degree polynomials have the same behavior on both ends:

• Positive leading coefficient: both ends rise (e.g., $f(x) = x^2$ or $f(x) = x^4$)
• Negative leading coefficient: both ends fall (e.g., $f(x) = -x^2$ or $f(x) = -x^4$)

Odd-degree polynomials have opposite behavior on each end:

• Positive leading coefficient: falls left, rises right (e.g., $f(x) = x^3$)
• Negative leading coefficient: rises left, falls right (e.g., $f(x) = -x^3$)

A quick way to remember: for odd-degree with a positive leading coefficient, the graph goes from lower-left to upper-right, like a stretched-out "S."

Degree and Graph Shape

Relationship Between Degree and Turning Points

A polynomial of degree $n$ has at most $n - 1$ turning points (local maxima and minima). "At most" matters here: a polynomial doesn't always use all of them.

• Degree 2 (quadratic): at most 1 turning point
• Degree 3 (cubic): at most 2 turning points
• Degree 4 (quartic): at most 3 turning points

As the degree increases, the graph can wiggle more, creating additional peaks and valleys.

A Note on Symmetry

You may notice that certain simple polynomials have symmetry. For instance, $f(x) = x^2$ is symmetric about the y-axis, and $f(x) = x^3$ is symmetric about the origin. However, this symmetry only holds for polynomials that contain exclusively even-powered terms (for y-axis symmetry) or exclusively odd-powered terms (for origin symmetry). A function like $f(x) = x^3 + x^2$ has no symmetry at all, even though it's odd-degree. So don't assume symmetry based on degree alone; check the actual terms.

Finding Zeros of Polynomials

Identifying Zeros Graphically and Algebraically

Zeros (also called roots) are the x-values where $f(x) = 0$. On the graph, these are the x-intercepts.

To find zeros algebraically:

1. Factor the polynomial completely.
2. Set each factor equal to zero and solve (zero product property).
3. If factoring isn't straightforward, use synthetic division or long division to test potential rational zeros, then factor the resulting quotient.

Graphically, zeros appear where the curve crosses or touches the x-axis. A graphing calculator can help you estimate zeros that are hard to find by hand.

Zero Multiplicity and Graph Behavior

The multiplicity of a zero is how many times that factor appears in the factored form. Multiplicity controls what the graph does at that x-intercept:

• Odd multiplicity (1, 3, 5, ...): the graph crosses the x-axis at that zero.
• Even multiplicity (2, 4, 6, ...): the graph touches the x-axis and turns back, without crossing.

For example, in $f(x) = (x + 3)^2(x - 1)$:

• $x = -3$ has multiplicity 2 (even), so the graph touches the axis at $(-3, 0)$ and bounces back.
• $x = 1$ has multiplicity 1 (odd), so the graph crosses through $(1, 0)$.

Higher multiplicities create flatter contact with the axis. A multiplicity-2 zero looks like a gentle bounce, while a multiplicity-3 zero crosses but with a flattened "inflection" shape.

Graphing Polynomial Functions

Sketching Process

Follow these steps to sketch a polynomial by hand:

1. Identify the degree and leading coefficient. This tells you the end behavior and the maximum number of turning points.
2. Find the zeros and their multiplicities. Factor the polynomial or use division techniques. Determine whether the graph crosses or touches at each zero.
3. Find the y-intercept. Plug in $x = 0$ to get $f(0)$.
4. Plot the intercepts on your axes.
5. Sketch the curve connecting the intercepts, making sure the graph:
   • Exhibits the correct end behavior
   • Crosses or bounces at each zero according to its multiplicity
   • Doesn't exceed the maximum number of turning points

Worked Example

Graph $f(x) = (x - 1)(x + 2)^2$.

1. Degree and leading coefficient: Expanding gives a leading term of $x^3$, so degree 3 with a positive leading coefficient. End behavior: falls left, rises right. At most 2 turning points.

2. Zeros and multiplicities:

   • $x = 1$, multiplicity 1 (odd) → graph crosses at $(1, 0)$
   • $x = -2$, multiplicity 2 (even) → graph touches and bounces at $(-2, 0)$

3. Y-intercept: $f(0) = (0 - 1)(0 + 2)^2 = (-1)(4) = -4$, so the graph passes through $(0, -4)$.

4. Sketch: Starting from the lower left (falling), the curve rises to touch the x-axis at $(-2, 0)$, dips down through $(0, -4)$, then crosses the x-axis at $(1, 0)$ and continues rising to the upper right.

Plotting a few extra points between zeros can help you refine the shape, but these key features give you a solid sketch.