Average rate of change on [a, b] = (f(b) − f(a)) / (b − a). It’s the slope of the secant line through (a, f(a)) and (b, f(b)) and has units = (change in output)/(change in input). For example, for f(x)=x^2 on [1,3], AROC = (9−1)/(3−1)=8/2=4. Key AP things to remember from Topic 1.3: - Linear functions: AROC is the same over any interval (constant slope) (CED 1.3.A.1). - Quadratics: AROC over consecutive equal-length intervals changes linearly; second differences constant (CED 1.3.A.2, 1.3.B.2). - Concavity: If AROC over small equal intervals is increasing → concave up; decreasing → concave down (CED 1.3.B.3). On the exam show work (write the secant-slope calculation), include units, and if given a table use first/second differences or the difference quotient. For the topic study guide and extra practice, see the Topic 1.3 guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

The average rate of change of a function f over the closed interval [a, b] is the change in output divided by the change in input—i.e. the slope of the secant line between (a, f(a)) and (b, f(b)). Formula: Average rate of change = (f(b) − f(a)) / (b − a). Notes tied to the CED: for linear functions this value is constant for any interval (1.3.A.1); for quadratics the average rates over consecutive equal-length intervals form a linear function (1.3.A.2). On the AP exam you’ll often interpret this as slope of a secant and use it to reason about concavity and changes in rate (1.3.A.3, 1.3.B). For a quick topic review, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA). For extra practice problems, check (https://library.fiveable.me/practice/ap-pre-calculus).

Use the average rate of change when you want the overall change over an interval [a, b]. Compute it as (f(b) − f(a)) / (b − a); that’s the slope of the secant line between (a, f(a)) and (b, f(b)) (CED 1.3.A.3). For example, average speed or the change in test scores across a month uses this. Use the instantaneous rate of change when you want how fast the function is changing at a single input value x = a. In precalc you usually approximate it by taking average rates over smaller and smaller intervals around a (the slope of secant lines approaching a)—conceptually the slope of the tangent line. For linear functions the average and instantaneous rates are the same everywhere (constant slope). For quadratics, average rates over equal-length intervals change linearly and instantaneous rate varies with x (first differences and second differences tell you this; CED 1.3.A.1–1.3.B.2). On the AP exam you’ll be asked to compute average rates (secant slopes) and reason about how those averages change (concavity, second differences). For a quick topic review see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA). For more practice problems, try the AP Precalculus practice bank (https://library.fiveable.me/practice/ap-pre-calculus).

Short answer: linear functions have a constant average rate of change; quadratics do not—their average rates of change over equal-length intervals change linearly. Why that matters (CED language): For a linear function, the average rate of change over any interval equals the slope so it’s the same everywhere (1.3.A.1). For a quadratic f(x)=ax^2+bx+c, the average rates of change over consecutive equal-length x-intervals form a linear function—so those average rates change at a constant rate (1.3.A.2 and 1.3.B.2). In difference language: linear functions have constant first differences; quadratics have constant second differences. Geometrically, the average rate of change on [a,b] is the slope of the secant line between (a,f(a)) and (b,f(b)) (1.3.A.3). If those average rates increase for small intervals the graph is concave up; if they decrease it’s concave down (1.3.B.3). Want practice? Review Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and the Unit 1 overview (https://library.fiveable.me/ap-pre-calculus/unit-1). For lots of practice problems go here (https://library.fiveable.me/practice/ap-pre-calculus).

Think of a linear function f(x) = mx + b. The average rate of change on [a, b] is (f(b) − f(a)) / (b − a). Plug in f: (f(b) − f(a)) / (b − a) = (m b + b0 − (m a + b0)) / (b − a) = m(b − a)/(b − a) = m. So no matter which a and b you pick, the difference quotient simplifies to the same number m—the slope. That’s why all secant lines between any two points on a line have the same slope, and first differences over equal-length intervals are constant. In CED terms this is Essential Knowledge 1.3.A.1 and 1.3.B.1: the average rates of change form a constant function (so their change is zero). If you want practice recognizing this from tables and graphs (first differences, secant slopes), check the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

Step-by-step to get the slope of a secant line (AKA the average rate of change on [a,b]): 1. Identify the interval [a, b]. Make sure a ≠ b. 2. Compute the function values f(a) and f(b). 3. Use the difference quotient: slope = (f(b) − f(a)) / (b − a). That’s the rise over run. 4. Simplify or approximate (keep 3 decimal places on AP Part A if using a calculator). 5. Interpret the result with units (e.g., miles/hour if f gives miles and x is hours). Quick example: f(x)=x^2 on [1,3]. f(1)=1, f(3)=9 so slope = (9−1)/(3−1)=8/2=4. That means the average rate of change is 4 units per x-unit. Remember: on the AP this slope is exactly the secant-line slope from (a,f(a)) to (b,f(b)) (CED 1.3.A.3). For more practice and quick review, check the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) or the Unit 1 overview (https://library.fiveable.me/ap-pre-calculus/unit-1).

Saying “quadratic functions have average rates of change that form a linear pattern” means this: if you take f(x)=ax^2+bx+c and compute the average rate of change (slope of the secant) over consecutive equal-length x-intervals, those slopes change by the same amount each step—they lie on a line. Practically, compute f(x+h)-f(x) over h; for a quadratic that quotient simplifies to a linear expression in x (and h), so the first differences are linear and the second differences are constant (2a). That constant second difference = the constant rate at which the average rates themselves change (CED 1.3.A.2 and 1.3.B.2). On the exam, average rate = secant slope on [a,b] (CED 1.3.A.3); noticing linear first differences helps you identify quadratics and reason about concavity (increasing average rates → concave up). For more examples and practice, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

Look at how the average rate of change (slope of secant lines) behaves on equal small x-intervals. - If those average rates of change are increasing as x increases (each successive secant slope is larger), the graph is concave up. - If the average rates of change are decreasing, the graph is concave down. Practical checks: - For a quadratic f(x)=ax^2+bx+c, the average rates over equal-length intervals form a linear function. The second difference (or the coefficient 2a) tells you concavity: if 2a (or the second differences) > 0, concave up; if < 0, concave down. - For linear functions the average rate is constant, so concavity is neither (rate of change isn’t changing). This is exactly what the CED emphasizes: concavity comes from whether the average rate of change is increasing or decreasing (1.3.B.3). For more examples and practice, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and extra practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

“Consecutive equal-length intervals” just means you look at back-to-back input intervals that all have the same width (same Δx). For example, intervals [0,1], [1,2], [2,3] are consecutive equal-length intervals because each has length 1. In the CED language this matters because average rate of change = (change in output)/(change in input) over [a, b]—if every interval has the same b−a, you can compare those average rates directly. Why it’s useful: for a linear function the average rate of change over any equal-length interval is the same (constant). For a quadratic, the average rates over consecutive equal-length intervals form a linear sequence. Example: f(x)=x^2, Δx=1: - [0,1]: slope = (1−0)/1 = 1 - [1,2]: slope = (4−1)/1 = 3 - [2,3]: slope = (9−4)/1 = 5 Those slopes (1,3,5) change at a constant rate (second difference constant), showing the quadratic’s average rates are linear. For more practice, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) or try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

Because a linear function is f(x) = mx + b, every secant (and tangent) slope equals m. The average rate of change over any equal-length input interval is always m (constant), so when you look at how those average rates themselves change from one interval to the next, there is no change—the difference is 0. In CED terms: average rates for a linear function are given by a constant function, so the change in average rates is zero (1.3.A.1 and 1.3.B.1). Think in finite-differences language: first differences for equal x-steps are all the same for a line, so second differences = 0. That zero second difference is the formal way to say “rate of change changing at zero.” This is tested on the AP build-up to understanding concavity and quadratic vs linear behavior (Topic 1.3). For a quick review, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and drill practice at (https://library.fiveable.me/practice/ap-pre-calculus).

A secant line is just the line through two points on a function’s graph, (a, f(a)) and (b, f(b)). Its slope = (f(b) − f(a)) / (b − a), which is exactly the average rate of change of f on [a, b] (CED 1.3.A.3). So “secant line” and “average rate of change” are two ways to talk about the same idea: the secant’s slope measures how much the output changes per unit input, on average, over that interval. Use the connection to spot behavior quickly: for a linear function every secant has the same slope (average rate is constant, 1.3.A.1). For a quadratic, slopes of secants over equal-length intervals change linearly (1.3.A.2), and how those slopes increase or decrease tells you concavity—if average rates over small equal intervals increase, the graph is concave up; if they decrease, concave down (1.3.B.3). On the exam, you’ll often compute that difference quotient or interpret secant slopes from graphs or tables—practice both (study guide: https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA; unit overview: https://library.fiveable.me/ap-pre-calculus/unit-1).

Think of average rate of change as the slope of a secant line over an interval [a, b]. If those secant slopes over consecutive equal small intervals are getting larger as x increases, the graph is concave up; if the secant slopes are getting smaller, the graph is concave down (CED 1.3.B.3). Quick rules you can use: - Linear: average rate of change is constant (no change → concavity not applicable; second difference = 0) (CED 1.3.A.1, 1.3.B.1). - Quadratic: average rates over equal-length intervals form a linear function, so the change in those average rates is constant (second differences constant). Example: f(x)=x^2 → secant slopes increase (concave up); f(x)=-x^2 → secant slopes decrease (concave down) (CED 1.3.A.2, 1.3.B.2). On the AP exam you’ll be asked to justify concavity by describing whether average rates of change (secant slopes) over small equal intervals are increasing or decreasing—use first and second differences or compare secant slopes. For a refresher, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA). For extra practice, try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

Use the secant-slope idea: average rate of change (AROC) on [a,b] = (f(b) − f(a)) / (b − a). Steps: 1) identify a and b from the word problem (include units), 2) evaluate f at those inputs, 3) compute the quotient and attach units (units of f per unit of x). Example: if f(2)=5 and f(5)=17, AROC = (17−5)/(5−2) = 12/3 = 4 units per input unit. Quick tips from the CED: for linear functions AROC is constant (so any interval gives same answer). For quadratics, AROC over equal-length consecutive intervals changes linearly—use first and second differences to spot patterns and concavity (increasing AROC → concave up; decreasing → concave down). On the AP exam show all steps, include units, and when in Part A use your graphing calculator to check values (calculator allowed there). For more worked examples and exam-style practice, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA), the Unit 1 overview (https://library.fiveable.me/ap-pre-calculus/unit-1), and thousands of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Think of two different “rates”: - The rate a quadratic function changes on an interval is its average rate of change = [f(b) − f(a)]/(b − a). For consecutive equal-length intervals (like [x, x+1]) those average rates form a linear function. Example: f(x)=ax^2+bx+c gives average over [x,x+1] equal to 2ax + a + b—linear in x with slope 2a. - The rate the average rate of change itself changes is the change of those average-rate values (the “rate of the rate”). For the quadratic above that change between consecutive equal-length intervals is constant (2a for unit intervals). In discrete terms: first differences of f are linear; second differences are constant. For a linear function those second differences are zero (average rates don’t change). Why it matters for AP: connect secant slopes, first/second differences, and concavity (increasing average rates ⇒ concave up). For more examples and practice, see the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and extra problems (https://library.fiveable.me/practice/ap-pre-calculus).

Short answer: no—just know the definitions and how to compute them, not a bunch of special “magic” formulas. What to memorize (few, useful things) - Average rate of change on [a,b]: (f(b) − f(a)) / (b − a). This is the slope of the secant line (CED 1.3.A.3). - For linear functions: that average rate is constant over any equal-length interval (CED 1.3.A.1 → change in average rates = 0). - For quadratics: first differences over equal x-steps form a linear sequence and second differences are constant (CED 1.3.A.2, 1.3.B.2). How you’ll use it on the AP - Compute the difference quotient for MCQs/FRQs and interpret units or concavity (concave up if average rates increase; concave down if they decrease—CED 1.3.B.3). - Show your work on FRQs (write the secant slope or finite differences)—exam instructions require supporting work. Want practice? Use the Topic 1.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA) and the unit practice bank (https://library.fiveable.me/practice/ap-pre-calculus).