Graphing Polynomials and Intercept Behavior

The value of the exponent $p$ determines the shape of the graph near the intercept $(h, 0)$:

1. Multiplicity of 1 ($p=1$): The graph passes straight through the $x$-axis like a line.
2. Even Multiplicity ($p=2, 4, \dots$): The graph touches the $x$-axis and "bounces" back in the direction it came from, behaving like a parabola at that point.
3. Odd Multiplicity greater than 1 ($p=3, 5, \dots$): The graph passes through the $x$-axis but "flattens out" as it crosses, creating an S-shaped inflection point.

Exam Tip: When looking at a factored equation, always check the exponents on each factor. These exponents are your multiplicities and will tell you exactly whether to "cross" or "bounce" at each intercept.

Polynomial Sketching: A Step-by-Step Guide

To create a complete sketch of a polynomial, you should combine several pieces of information into a single cohesive shape. Follow this checklist:

1. End Behavior (Long Run Behavior): Determine where the graph goes as $x$ approaches infinity or negative infinity.
2. Turning Points: Note that a polynomial of degree $n$ can have at most $n-1$ turning points.
3. $y$-intercept: Locate the point $(0, y)$.
4. $x$-intercepts and Multiplicities: Plot the zeros and decide if the graph crosses or bounces at each.

Worked Example: Building a Polynomial from Scratch

Sometimes, you are asked to do the reverse: write an equation based on a given graph or set of characteristics.

Problem: Find a polynomial of degree 4 with:

• $x$-intercepts at $(2,0)$ with multiplicity 2.
• $x$-intercepts at $(-1,0)$ and $(-3,0)$ with multiplicity 1.
• A $y$-intercept at $(0,4)$.

Step 1: Set up the factored form Using the intercepts and multiplicities, we write the factors. Remember that for an intercept $h$, the factor is $(x - h)$.

• For $x = 2$ (mult 2): $(x - 2)^2$
• For $x = -1$ (mult 1): $(x - (-1)) \rightarrow (x + 1)$
• For $x = -3$ (mult 1): $(x - (-3)) \rightarrow (x + 3)$

We include a leading coefficient $a$ because we don't yet know the vertical stretch: $f(x) = a(x - 2)^2(x + 1)(x + 3)$

Step 2: Solve for $a$ using the $y$-intercept We know that when $x = 0$, $f(x) = 4$. Plug these values into our equation:

$$\begin{aligned} 4 &= a(0 - 2)^2(0 + 1)(0 + 3) \\ 4 &= a(-2)^2(1)(3) \\ 4 &= a(4)(1)(3) \\ 4 &= 12a \end{aligned}$$

Solving for $a$: $a = \frac{4}{12} = \frac{1}{3}$

Final Equation: $f(x) = \frac{1}{3}(x - 2)^2(x + 1)(x + 3)$

By following these steps—identifying intercepts, determining multiplicity, and checking end behavior—you can master the art of sketching any polynomial function.