Graphs of Polynomial Functions

Polynomial graphs are smooth, continuous curves without any sharp corners or breaks. To sketch them accurately without plotting dozens of points, we analyze three main features: end behavior, zeros (x-intercepts), and the multiplicity of those zeros.

End Behavior

The "end behavior" of a polynomial describes what happens to the graph as $x$ gets very large (approaches $\infty$) or very small (approaches $-\infty$). This is completely determined by the function's leading term $ax^n$.

• Even Degree, Positive $a$: Both ends point up ($\uparrow, \uparrow$).
• Even Degree, Negative $a$: Both ends point down ($\downarrow, \downarrow$).
• Odd Degree, Positive $a$: Left end points down, right end points up ($\downarrow, \uparrow$).
• Odd Degree, Negative $a$: Left end points up, right end points down ($\uparrow, \downarrow$).

Example: Determine the end behavior of $f(x) = -3x^5 + 2x^3 - x$.

• The leading term is $-3x^5$.
• The degree is $5$ (an odd number).
• The leading coefficient is $-3$ (a negative number).
• Conclusion: As $x \to -\infty$, $f(x) \to \infty$. As $x \to \infty$, $f(x) \to -\infty$. The graph rises on the left and falls on the right.

Zeros and Multiplicity

The zeros (or roots) of a polynomial are the $x$-values where $f(x) = 0$. These are the x-intercepts of the graph.

The multiplicity of a zero is the exponent of its corresponding factor $(x - c)^m$. The multiplicity tells us how the graph behaves at that specific x-intercept:

• Odd Multiplicity ($m = 1, 3, 5, ...$): The graph crosses the x-axis at $x = c$. (If $m > 1$, it flattens out slightly as it crosses).
• Even Multiplicity ($m = 2, 4, 6, ...$): The graph touches the x-axis and bounces back at $x = c$.

Sketching a Polynomial Graph

Let's put it all together to sketch a graph.

Example: Sketch the graph of $f(x) = -(x + 1)^2(x - 2)(x - 4)$.

Step 1: Determine the End Behavior Imagine expanding the polynomial to find the leading term. Multiply the leading terms of each factor: $-(x)^2(x)(x) = -x^4$.

• The degree is $4$ (even) and the leading coefficient is negative.
• End behavior: Both ends point down ($\downarrow, \downarrow$).

Step 2: Find the Zeros and their Multiplicity Set each factor to zero:

• $(x + 1)^2 = 0 \implies x = -1$. Multiplicity is $2$ (even), so the graph touches the x-axis.
• $x - 2 = 0 \implies x = 2$. Multiplicity is $1$ (odd), so the graph crosses the x-axis.
• $x - 4 = 0 \implies x = 4$. Multiplicity is $1$ (odd), so the graph crosses the x-axis.

Step 3: Find the Y-intercept Set $x = 0$: $f(0) = -(0 + 1)^2(0 - 2)(0 - 4) = -(1)(-2)(-4) = -8$ The y-intercept is $(0, -8)$.

Step 4: Draw the Sketch

1. Start from the bottom left (since the left end points down).
2. Bring the curve up to touch the x-axis at $x = -1$, then bounce back down.
3. Pass through the y-intercept at $(0, -8)$.
4. Curve back up to cross the x-axis at $x = 2$.
5. Curve back down to cross the x-axis at $x = 4$.
6. Continue downwards towards the bottom right (matching the end behavior).