AP PRECALCULUS | UNIT 1: POLYNOMIAL AND RATIONAL FUNCTIONS
1.2 Rates of Change
Master the concept of rates of change for AP Precalculus. These study notes cover average and instantaneous rates of change, the difference quotient, secant and tangent lines, interpreting rate of change from tables and graphs, positive and negative rates, increasing and decreasing functions, and concavity. All formulas are rendered in proper mathematical notation with worked examples and an interactive quiz.
Big Idea
A rate of change describes how one quantity changes relative to another. In AP Precalculus, rates of change connect the input-output behaviour of functions to slopes of lines and curves. Understanding rates of change is the conceptual bridge from precalculus to calculus and appears throughout every unit of the AP exam.
Average Rate of Change (AROC)
The average rate of change of a function \( f \) over the interval \([a, b]\) measures how much the output changes per unit change in input across that interval. Geometrically, it is the slope of the secant line connecting the two points \((a, f(a))\) and \((b, f(b))\) on the graph of \( f \).
Average Rate of Change Formula
\[ \text{AROC} = \frac{f(b) - f(a)}{b - a} = \frac{\Delta y}{\Delta x} \]
This is identical to the slope formula from algebra. The numerator is the change in output (\(\Delta y\)); the denominator is the change in input (\(\Delta x\)).
Worked Example 1 — AROC from a Function Rule
Find the average rate of change of \( f(x) = x^2 - 3x + 5 \) over the interval \([1, 4]\).
Step 1: Calculate \( f(1) = 1 - 3 + 5 = 3 \)
Step 2: Calculate \( f(4) = 16 - 12 + 5 = 9 \)
Step 3: Apply the formula:
\[ \text{AROC} = \frac{f(4) - f(1)}{4 - 1} = \frac{9 - 3}{3} = \frac{6}{3} = 2 \]
Interpretation: On average, the output increases by 2 units for every 1-unit increase in input over \([1, 4]\).
Worked Example 2 — AROC from a Table
Given the table below, find the AROC of \( g(x) \) over \([2, 6]\):
| \( x \) | \( g(x) \) |
|---|---|
| 2 | 10 |
| 4 | 18 |
| 6 | 30 |
\[ \text{AROC} = \frac{g(6) - g(2)}{6 - 2} = \frac{30 - 10}{4} = \frac{20}{4} = 5 \]
Interpretation: The output of \( g \) increases by an average of 5 units per unit of input over \([2, 6]\).
Secant Lines and Average Rate of Change
A secant line is a line that intersects a curve at two points. The slope of the secant line through \((a, f(a))\) and \((b, f(b))\) equals the average rate of change of \( f \) on \([a, b]\).
Equation of a Secant Line
\[ y - f(a) = \left(\frac{f(b) - f(a)}{b - a}\right)(x - a) \]
This is point-slope form where the slope is the AROC. You can use either endpoint as the point.
Key Insight for the AP Exam
The secant line gives a linear approximation of the function's behaviour between two points. As the interval narrows (i.e. \( b \to a \)), the secant line approaches the tangent line and the AROC approaches the instantaneous rate of change. This idea is the foundation of calculus.
The Difference Quotient
The difference quotient is a general formula for the average rate of change of a function \( f \) over an interval of width \( h \) starting at \( x \). It is the algebraic form that connects the AROC to the derivative in calculus.
Difference Quotient Formula
\[ \frac{f(x + h) - f(x)}{h}, \quad h \neq 0 \]
Here \( h \) is the change in \( x \) (the run), and \( f(x+h) - f(x) \) is the change in \( y \) (the rise). This is the AROC over the interval \([x, x+h]\).
Worked Example 3 — Computing the Difference Quotient
Find and simplify the difference quotient for \( f(x) = 3x^2 + 2x \).
Step 1: Compute \( f(x + h) \):
\[ f(x+h) = 3(x+h)^2 + 2(x+h) = 3x^2 + 6xh + 3h^2 + 2x + 2h \]
Step 2: Compute \( f(x+h) - f(x) \):
\[ f(x+h) - f(x) = 6xh + 3h^2 + 2h \]
Step 3: Divide by \( h \):
\[ \frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 2h}{h} = 6x + 3h + 2 \]
Result: \( 6x + 3h + 2 \). As \( h \to 0 \), this approaches \( 6x + 2 \), which is the derivative \( f'(x) \). In AP Precalculus, you simplify the difference quotient but do not formally take the limit.
Instantaneous Rate of Change (Preview)
While AP Precalculus focuses on average rates of change, understanding the concept of instantaneous rate of change is essential for building towards calculus. The instantaneous rate of change at a point \( x = a \) is the slope of the tangent line to the curve at that point.
Instantaneous Rate of Change (Conceptual)
\[ \text{IROC at } x = a = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
In precalculus, you estimate the instantaneous rate of change by computing the AROC over smaller and smaller intervals around \( a \).
Average Rate of Change
- Measured over an interval \([a, b]\)
- Equals the slope of the secant line
- Uses the formula \(\frac{f(b)-f(a)}{b-a}\)
- Gives a single, fixed value for the interval
Instantaneous Rate of Change
- Measured at a single point
- Equals the slope of the tangent line
- Found by taking a limit as \( h \to 0 \)
- Changes from point to point along a curve
Rates of Change from Tables
On the AP Precalculus exam, you will often need to compute and compare rates of change using tabular data. The AROC between consecutive entries reveals how the function's behaviour changes across each sub-interval.
Worked Example 4 — Analysing AROC from a Table
| \( x \) | \( f(x) \) | AROC on sub-interval |
|---|---|---|
| 0 | 2 | — |
| 1 | 5 | \(\frac{5-2}{1-0} = 3\) |
| 2 | 10 | \(\frac{10-5}{2-1} = 5\) |
| 3 | 17 | \(\frac{17-10}{3-2} = 7\) |
| 4 | 26 | \(\frac{26-17}{4-3} = 9\) |
Observation: The AROC is increasing (3, 5, 7, 9). Since the rate of change itself is growing, the function is increasing at a faster and faster rate. This tells us the function is concave up.
Second Differences: \( 5 - 3 = 2,\ 7 - 5 = 2,\ 9 - 7 = 2 \). Constant second differences confirm a quadratic function.
Positive, Negative, and Zero Rates of Change
The sign of the rate of change tells you whether the function is increasing, decreasing, or momentarily flat on the given interval.
Positive AROC
\( f(b) > f(a) \)
The function is increasing on \([a, b]\) on average.
The secant line slopes upward from left to right.
Negative AROC
\( f(b) < f(a) \)
The function is decreasing on \([a, b]\) on average.
The secant line slopes downward from left to right.
Zero AROC
\( f(b) = f(a) \)
The output is the same at both endpoints (the function may still change between them).
The secant line is horizontal.
Caution: Zero AROC does not mean constant
A zero average rate of change over \([a, b]\) only means that \( f(a) = f(b) \). The function could increase and then decrease (or vice versa) between the endpoints. For example, \( f(x) = x^2 \) has AROC = 0 over \([-2, 2]\) even though \( f \) decreases then increases on that interval.
Increasing and Decreasing Functions
A function is classified as increasing or decreasing based on the sign of its rate of change. This classification is crucial for describing function behaviour on the AP exam.
Formal Definitions
Increasing on \((a, b)\): For all \( x_1, x_2 \in (a, b) \), if \( x_1 < x_2 \) then \( f(x_1) < f(x_2) \).
Decreasing on \((a, b)\): For all \( x_1, x_2 \in (a, b) \), if \( x_1 < x_2 \) then \( f(x_1)> f(x_2) \).
In AP Precalculus, use open intervals for increasing/decreasing and report turning points (local extrema) separately.
Increasing
As input increases, output increases. AROC > 0 on every sub-interval. Graph goes up from left to right.
Decreasing
As input increases, output decreases. AROC < 0 on every sub-interval. Graph goes down from left to right.
Concavity and the Rate of Change of the Rate of Change
Concavity describes how the rate of change itself is changing. If the AROC is increasing over successive intervals, the graph is concave up. If the AROC is decreasing over successive intervals, the graph is concave down.
Concave Up
- AROC is increasing over successive intervals
- Graph curves like a cup (opens upward)
- Function is increasing at an increasing rate, or decreasing at a decreasing rate
- Tangent lines lie below the curve
Concave Down
- AROC is decreasing over successive intervals
- Graph curves like a frown (opens downward)
- Function is increasing at a decreasing rate, or decreasing at an increasing rate
- Tangent lines lie above the curve
Detecting Concavity from a Table
Step 1: Compute the AROC on each consecutive sub-interval (first differences of \( f \) divided by \(\Delta x\)).
Step 2: Check whether those AROC values are increasing or decreasing.
Step 3: If AROC values increase → concave up. If they decrease → concave down. If constant → linear (no concavity).
Worked Example 5 — Concavity from Table Analysis
| \( x \) | \( h(x) \) | AROC on sub-interval |
|---|---|---|
| 0 | 100 | — |
| 1 | 90 | \(\frac{90-100}{1} = -10\) |
| 2 | 75 | \(\frac{75-90}{1} = -15\) |
| 3 | 55 | \(\frac{55-75}{1} = -20\) |
Analysis: The AROC values are \(-10, -15, -20\), which are decreasing (becoming more negative). The function is decreasing (negative AROC) and concave down (AROC is decreasing). In context: the function is decreasing at an increasing rate.
Rate of Change of Linear Functions
A linear function has a constant rate of change. The AROC is the same over every interval, and it equals the slope \( m \) of the line.
Linear Function Standard Form
\[ f(x) = mx + b \]
\[ \text{AROC over any interval} = m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \quad \text{(constant)} \]
Identifying Linear Functions from Data
Given a table of values with equally spaced input values, if the first differences (successive output changes) are constant, the function is linear. The constant first difference divided by the constant input spacing equals the slope.
Rates of Change and Function Types
The behaviour of the rate of change reveals the type of function. This is a powerful tool for identifying and classifying functions on the AP exam.
| Function Type | Rate of Change Behaviour | Concavity | Key Signature |
|---|---|---|---|
| Linear: \( f(x) = mx + b \) | Constant (equal to \( m \)) | None (straight line) | First differences are constant |
| Quadratic: \( f(x) = ax^2 + bx + c \) | Linear (changes at a constant rate) | Up if \( a > 0 \), down if \( a < 0 \) | Second differences are constant |
| Exponential: \( f(x) = ab^x \) | Proportional to the function value itself | Always concave up (if \( a > 0 \)) | Successive ratios are constant |
| Polynomial (degree \( n \)) | \( n \)-th differences are constant | Changes direction at most \( n-2 \) times | \( n \)-th differences constant |
Key Relationship for Quadratics
For \( f(x) = ax^2 + bx + c \) over equally spaced intervals with step size \(\Delta x\):
\[ \text{Second Difference} = 2a(\Delta x)^2 \]
If \(\Delta x = 1\), the second difference = \(2a\). This lets you identify the leading coefficient from a table.
Rates of Change in Context (Real-World Applications)
Rate of change has direct real-world meaning. Interpreting it correctly in context is tested on both the multiple-choice and free-response sections of the AP exam.
Distance & Time
\(\frac{\Delta d}{\Delta t}\) = average speed/velocity. Units: metres per second (m/s), miles per hour (mph).
Population & Time
\(\frac{\Delta P}{\Delta t}\) = average growth rate. Units: people per year.
Revenue & Quantity
\(\frac{\Delta R}{\Delta q}\) = marginal revenue. Units: dollars per unit.
Temperature & Altitude
\(\frac{\Delta T}{\Delta h}\) = lapse rate. Units: degrees per metre (or per km).
AP Exam Tip: Interpreting in Context
When the AP exam asks you to "interpret the rate of change in context", your answer must include three things: (1) a numerical value, (2) correct units, and (3) a contextual sentence stating what the value means in the given scenario. For example: "From 2010 to 2015, the population increased by an average of 1,200 people per year."
Common AP Exam Mistakes
- Confusing slope with the value of the function: The rate of change is \(\frac{\Delta y}{\Delta x}\), not simply \( f(a) \) or \( f(b) \). The output values are inputs to the formula, not the answer.
- Swapping numerator and denominator: The rate of change is \(\frac{f(b)-f(a)}{b-a}\), not \(\frac{b-a}{f(b)-f(a)}\). Always put the output change on top, the input change on the bottom.
- Assuming zero AROC means the function is constant: A zero average rate of change only means \( f(a) = f(b) \). The function can fluctuate between the endpoints. Always check sub-interval behaviour.
- Ignoring units in context problems: When a problem says "the population was 5,000 in 2010 and 8,000 in 2015," your answer must include "people per year." Unitless numerical answers lose points on free-response questions.
- Confusing concavity with increasing/decreasing: Concavity describes whether the rate of change is increasing or decreasing. A function can be both increasing and concave down (e.g. \( f(x) = \sqrt{x} \) for \( x > 0 \)).
- Forgetting to simplify the difference quotient: Always cancel the \( h \) factor from the numerator. Unsimplified answers are not considered complete.
Official and Recommended Resources
The following are verified official and authoritative resources for AP Precalculus 1.2 Rates of Change.
College Board — AP Precalculus
Official course description, exam format, scoring guidelines, and sample questions for AP Precalculus.
apcentral.collegeboard.org — AP PrecalculusAP Classroom
AP Classroom provides topic questions, progress checks, and personal progress dashboards. Access through your College Board account.
myap.collegeboard.orgKhan Academy — Average Rate of Change
Free video lessons and practice exercises on average rate of change, difference quotients, and secant lines.
khanacademy.org — Average Rate of ChangeDesmos Graphing Calculator
Free online graphing calculator to visualise secant lines, tangent lines, and rate of change. Allowed on the AP exam.
desmos.com/calculatorPaul's Online Math Notes
Exceptionally clear notes on rates of change, tangent lines, and the transition from precalculus to calculus concepts.
tutorial.math.lamar.edu — Rate of ChangeMIT OpenCourseWare — Single Variable Calculus
University-level lecture notes and videos on rates of change and the derivative, building directly from precalculus concepts.
ocw.mit.edu — Single Variable CalculusTest Your Knowledge: 1.2 Rates of Change Quiz
Check your understanding of the key concepts. Select the best answer for each question.
Key Takeaways for the AP Exam
- The average rate of change (AROC) is \(\frac{f(b)-f(a)}{b-a}\) and equals the slope of the secant line through \((a, f(a))\) and \((b, f(b))\).
- The difference quotient \(\frac{f(x+h)-f(x)}{h}\) generalises the AROC and bridges precalculus to calculus.
- Positive AROC means the function increases on average; negative AROC means it decreases on average; zero AROC does not mean constant.
- Concavity is determined by whether the AROC is increasing (concave up) or decreasing (concave down) over successive sub-intervals.
- Constant first differences indicate a linear function; constant second differences indicate a quadratic function; constant ratios indicate an exponential function.
- Always interpret rate of change with a numerical value, correct units, and a contextual sentence on the AP exam.
Frequently Asked Questions About AP Precalculus 1.2
What is the difference between average rate of change and instantaneous rate of change?
Average rate of change is measured over an interval \([a, b]\) and equals the slope of the secant line connecting two points on the graph. Instantaneous rate of change is measured at a single point and equals the slope of the tangent line. In precalculus, you compute average rates of change and estimate instantaneous rates by making the interval very small. The formal instantaneous rate requires a limit (calculus).
How do you find the rate of change from a table?
Select two rows from the table. Subtract the output values (numerator) and subtract the input values (denominator). The result \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\) is the average rate of change over that sub-interval. Repeat for consecutive rows to see how the rate changes across the table, which reveals concavity and function type.
What does concavity have to do with rates of change?
Concavity describes how the rate of change itself is changing. If the AROC over consecutive sub-intervals is increasing, the graph is concave up (curves like a cup). If the AROC is decreasing, the graph is concave down (curves like a frown). This is essentially the "rate of change of the rate of change."
What is the difference quotient and why does it matter?
The difference quotient is \(\frac{f(x+h)-f(x)}{h}\). It gives the average rate of change over the interval \([x, x+h]\) in a general algebraic form. When you simplify it and then let \( h \to 0 \), you get the derivative. In AP Precalculus, simplifying the difference quotient builds the algebraic skills needed for calculus.
How do constant differences help identify function types?
For equally spaced input values: if the first differences (output changes) are constant, the function is linear. If the second differences (changes of the changes) are constant, it is quadratic. If the successive ratios of output values are constant, it is exponential. This method works reliably on tables and is a frequent AP exam technique.
Can a function be increasing and concave down at the same time?
Yes. A function can be increasing (positive rate of change) but concave down (the rate of change is decreasing). This means the function is still going up, but at a slowing pace. A classic example is \( f(x) = \sqrt{x} \) for \( x > 0 \): it always increases, but the slope gets flatter as \( x \) grows.
About the Author
Adam
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Expert in mathematics education with extensive experience teaching AP Precalculus, AP Calculus AB/BC, and IB Mathematics. Specialises in building conceptual understanding of rates of change, limits, and derivatives through clear worked examples and visual explanations.
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Start here (prerequisites)
Practice and application
- 1.3 Rates of Change in Linear and Quadratic Functions
- 1.4 Polynomial Functions and Rates of Change
- Slope Guide
- Slope-Intercept Formula
Next steps
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