AP Precalculus: Polynomials and Rates of Change Study Guide

Factors of a Function

A factor of a function $f(x)$ is an expression that, when multiplied by other factors, results in the original function. If $(x - c)$ is a factor of $f(x)$, then $f(c) = 0$, meaning the graph must cross or touch the x-axis at $x = c$.

In more complex problems, you might be given a factor like $(\frac{x}{2} - 4)$ and asked how it affects your knowledge of the degree $n$. Knowing a specific factor helps confirm the existence of a root, which can help you refine your estimate of the minimum degree required to create that specific curve.

Rates of Change

The rate of change (ROC) describes how much the output $y$ changes relative to the change in the input $x$. In Precalculus, we primarily focus on the Average Rate of Change (AROC) over a specific interval.

Calculating Average Rate of Change

To find the AROC on an interval $[x_1, x_2]$, use the slope formula: $\text{AROC} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}$

For example, if you are given a table where $g(0) = 3$ and $g(3) = 1$, the AROC on the interval $[0, 3]$ is: $\frac{1 - 3}{3 - 0} = \frac{-2}{3}$

Linear vs. Non-Linear Functions

If a function $g(x)$ is linear, its rate of change is constant. This means the slope between any two points in the table will be exactly the same. If you are asked to find unknown constants $a$ or $b$ in a linear function, you can set the slopes between different intervals equal to each other.

Concavity and Quadratic Functions

When a function is not linear, its rate of change changes. This brings us to the concept of concavity.

Concave Down Quadratic Functions

A quadratic function $f(x) = ax^2 + bx + c$ is concave down if the leading coefficient $a$ is negative. Visually, this looks like an upside-down "U" or a mountain shape.

In a concave down function, the rate of change is decreasing. This means that as you move from left to right, the slopes of the secant lines (the AROC) get smaller and smaller (becoming more negative).

Critical Comparison: If you are asked to compare values like $b$ or $3-a$ in a concave down context, remember that the "steepness" of the function is dropping. A concave down shape means the function is "slowing down" its ascent or "speeding up" its descent.

Summary Table for Quick Review