Welcome to 1.4: Polynomial Functions and Rates of Change!

In previous chapters, we looked at how linear functions change at a constant rate and how quadratic functions change at a linear rate. But what happens when we dive into "higher-degree" polynomials, like cubics (\(x^3\)) or quartics (\(x^4\))? In this chapter, we explore how the rate of change itself changes as we move along a polynomial curve. Think of this as the difference between driving a car at a steady speed (linear) versus a roller coaster that speeds up, slows down, and turns upside down!

1. The Basics: Average Rate of Change (AROC)

Before we look at the patterns, let’s quickly recap. The Average Rate of Change of a function \(f\) over an interval \([a, b]\) is the ratio of the change in output to the change in input. It’s essentially the slope of the line connecting two points on the graph.

Formula: \(AROC = \frac{f(b) - f(a)}{b - a}\)

Quick Review: If you remember Topic 1.2 and 1.3, you know that for a linear function, this value is always the same. For a polynomial of degree \(n \ge 3\), this value will change depending on which interval \([a, b]\) you choose.


2. Comparing Rates of Change by Degree

To understand polynomial rates of change, it helps to compare them to the "simpler" functions we already know:

  • Linear Functions (Degree 1): The rate of change is constant. It never speeds up or slows down.
  • Quadratic Functions (Degree 2): The rate of change is linear. It changes at a steady, constant rate.
  • Polynomial Functions (Degree \(n \ge 3\)): The rate of change is non-linear. It is represented by a polynomial of degree \(n-1\). This means the rate of change can increase, then decrease, then increase again!

Did you know? If you have a cubic function (degree 3), its rate of change behaves like a quadratic function (degree 2). If you have a quartic function (degree 4), its rate of change behaves like a cubic (degree 3)!


3. Increasing and Decreasing Rates of Change

One of the most important skills in AP Precalculus is describing how the rate is changing. We don't just care if the function \(f(x)\) is going up or down; we care if it is speeding up or slowing down in its growth.

The Rate of Change is Increasing

If the rate of change is increasing, the graph is "bending upward." In geometry terms, we call this concave up. Even if the function is decreasing (getting more negative), it can still have an increasing rate of change if it is becoming "less negative" (like going from a slope of \(-10\) to a slope of \(-2\)).

The Rate of Change is Decreasing

If the rate of change is decreasing, the graph is "bending downward." We call this concave down. The slopes of the tangent lines are getting smaller (e.g., from \(10\) to \(5\) to \(0\) to \(-5\)).

Key Takeaway:
- Rate of Change Increasing \(\implies\) Graph looks like a "cup" \(\cup\)
- Rate of Change Decreasing \(\implies\) Graph looks like a "frown" \(\cap\)


4. Change in Rates over Successive Intervals

In many AP questions, you will be given a table of values with equal-length input intervals (like \(x = 1, 2, 3, 4\)). To identify the behavior of a polynomial, you should look at the "differences of the differences":

  1. Calculate the First Differences (this is the change in \(y\) between steps). This tells you the rate of change.
  2. Calculate the Second Differences (this is the change in the first differences). This tells you if the rate of change is increasing or decreasing.

Example: If the first differences are \(2, 5, 9, 14\), the second differences are \(3, 4, 5\). Since the second differences are positive, the rate of change is increasing.


5. Common Pitfalls to Avoid

Don't confuse "Function is Increasing" with "Rate of Change is Increasing."
A function can be increasing while its rate of change is decreasing. Imagine a car that is still moving forward but is slamming on the brakes. The distance (the function) is still increasing, but the speed (the rate of change) is decreasing. On a graph, this looks like a curve that is rising but flattening out.

Watch the signs!
If the rate of change goes from \(-10\) to \(-20\), the rate of change is decreasing (because \(-20 < -10\)). If the rate of change goes from \(-20\) to \(-10\), the rate of change is increasing.


6. Summary and Practice Tips

Summary Table

If the Rate of Change is... | The Graph is... | Analogy
Positive & Increasing | Rising and Steeper | Rocket blasting off
Positive & Decreasing | Rising and Flatter | Car coasting to a stop
Negative & Increasing | Falling and Flatter | Biker reaching the bottom of a hill
Negative & Decreasing | Falling and Steeper | Ball dropped from a cliff

Key Practice Advice:

When you are asked to justify your answer on the AP Exam (Practice 3.C), always refer to the numerical values. Instead of saying "the graph looks steeper," say "The average rate of change over the interval \([2, 3]\) is \(5\), which is greater than the average rate of change over the interval \([1, 2]\), which is \(2\); therefore, the rate of change is increasing."

Note: For more on how these polynomials behave as \(x\) goes to infinity, check out Chapter 1.6: Polynomial Functions and End Behavior.