📏Honors Pre-Calculus Unit 1 Review

The average rate of change tells you how much a function's output changes, on average, per unit of input over a specific interval. Graphically, it's the slope of the secant line connecting two points on the function.

The formula is:

$\frac{f(b) - f(a)}{b - a}$

where $a$ and $b$ are the input values at each end of the interval.

How to calculate it:

Identify the interval $[a, b]$ you're working with.
Find $f(a)$ and $f(b)$ by plugging each endpoint into the function.
Subtract: $f(b) - f(a)$ gives you the total change in output.
Divide by $b - a$ to get the change per unit of input.

For example, if $f(x) = x^2$ on the interval $[1, 4]$ :

$\frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5$

So the function's output increases by an average of 5 units for every 1 unit increase in $x$ over that interval.

A few things to keep in mind:

A positive average rate of change means the function is generally increasing over that interval; a negative one means it's generally decreasing.
You can compare average rates of change across different intervals to see where a function is changing faster or slower.
As the interval gets smaller and smaller, the average rate of change approaches the instantaneous rate of change at a point. That's the idea behind derivatives, which you'll see in calculus.

Average rate of change calculation, Rates of Change and Behavior of Graphs · Precalculus

Intervals of Function Behavior

When you describe where a function is increasing, decreasing, or constant, you're describing its behavior over parts of its domain. You always state these intervals using the $x$ -values (input), not the $y$ -values.

Increasing: A function is increasing on an interval if the output goes up as the input goes up. Formally, $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in that interval. On the graph, the curve moves upward from left to right.

Decreasing: A function is decreasing on an interval if the output goes down as the input goes up. Formally, $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in that interval. The curve moves downward from left to right.

Constant: A function is constant on an interval if the output stays the same regardless of input. Formally, $f(x_1) = f(x_2)$ for all $x_1$ and $x_2$ in that interval. The graph is a horizontal segment.

When writing these intervals, use open intervals (parentheses, not brackets) because we describe behavior between the turning points, not at them. For instance, if a function has a peak at $x = 3$ , you might say it's increasing on $(-\infty, 3)$ and decreasing on $(3, \infty)$ .

Local vs. Absolute Extrema

Extrema are the high and low points of a function. The distinction between local and absolute is about scope: are you the tallest person in the room, or the tallest person in the world?

Local maximum: A point $(a, f(a))$ where $f(a)$ is greater than or equal to the function values at all nearby points. The function increases as you approach from the left and decreases as you move to the right. Think of it as the top of a hill.
Local minimum: A point $(a, f(a))$ where $f(a)$ is less than or equal to the function values at all nearby points. The function decreases as you approach from the left and increases as you move to the right. Think of it as the bottom of a valley.
Absolute (global) maximum: The single highest function value over the entire domain. A function might have several local maxima, but only one value can be the absolute maximum.
Absolute (global) minimum: The single lowest function value over the entire domain.

A few important notes:

An absolute extremum is also a local extremum, but a local extremum isn't necessarily absolute.
Not every function has absolute extrema. For example, $f(x) = x^3$ keeps increasing forever in both directions, so it has no absolute max or min.
On a closed interval $[a, b]$ , always check the endpoints as well as any turning points. Absolute extrema on a closed interval must occur at a local extremum or at an endpoint.

Applications of Graphical Analysis

Understanding rates of change and function behavior lets you solve practical problems by reading and interpreting graphs.

Optimization problems ask you to find the maximum or minimum value of a function in context. For example, a business might model profit as a function of units sold. The absolute maximum of that function tells you the production level that maximizes profit. You find it by identifying extrema.

Interpreting graphs in context means connecting the shape of a graph to what's actually happening. If a graph of a projectile's height is a downward-opening parabola, the vertex is the maximum height, the increasing interval is the ascent, and the decreasing interval is the descent. The average rate of change between two times gives the average velocity over that period.

Using average rate of change to compare options is straightforward: calculate the average rate of change for each option over the same interval, and the steeper rate tells you which is changing faster. If you're comparing two investments over 5 years, the one with the higher average rate of change in value gave a better average return over that period.

Additional Concepts in Graphical Analysis

These ideas go slightly beyond the core of this section, but they connect directly to what you're learning and will show up soon.

Tangent line: A line that touches the curve at exactly one point (locally) and has a slope equal to the instantaneous rate of change at that point. The secant line you use for average rate of change becomes a tangent line as the interval shrinks to zero.
Concavity: Describes how the curve bends. If the curve bends upward (like a bowl), it's concave up, and the rate of change is itself increasing. If it bends downward (like an upside-down bowl), it's concave down, and the rate of change is decreasing.
Inflection point: A point where the concavity switches from up to down or down to up. At an inflection point, the function changes how it's curving, which often signals a shift in the function's behavior.

📏Honors Pre-Calculus Unit 1 Review