Unit 1.1-1.3
1.1 Change in Tandem
Function
A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.
Input values = Domain = Independent variable (x)
Output values = Range = Dependent variable (y)
The dependent variable depends on the independent variable.
For example, if:
h = height of a football
t = seconds since the football was thrown
Then function notation is:
h(t)
Here, t is the input/independent variable, and h is the output/dependent variable.
Function Representations — VANG
Two variables vary in tandem according to a function rule.
A function can be represented in four ways:
VANG
V = Verbally
A = Analytically (equations)
N = Numerically (tables)
G = Graphically
Increasing Functions
A function is increasing on an interval when the output increases as the input increases (both increase.
Verbally:
input increases → output increases
Analytically:
If
a < b
then
f(a) < f(b)
Graphically:
The graph moves upward from left to right.
Decreasing Functions
A function is decreasing on an interval when the output decreases as the input increases.
Verbally:
input increases → output decreases
Analytically:
If
a < b
then
f(a) > f(b)
Graphically:
The graph moves downward from left to right.
Numerically — Using a Table
To determine whether a function is increasing or decreasing from a table:
Read the x-values from smallest to largest.
Watch what happens to f(x).
If f(x) consistently rises → increasing.
If f(x) consistently falls → decreasing.
For the notes example:
As x increases, f(x) increases, so the function is increasing.
Basic Elements of a Function's Graph
Zero
The graph intersects the x-axis when the output value is zero.
f(x) = 0
The corresponding input values are called zeros.
So:
Zero = x-value where f(x) = 0
Zeros correspond to x-intercepts.
y-intercept
The graph intersects the y-axis when the input value is zero.
x = 0
The output value is called the y-intercept:
f(0)
So the blanks in your sheet are:
The graph intersects the y-axis when the input value is zero. We call the output value the y-intercept.
Concavity
Concavity describes how the graph bends/how its rate of change changes, not simply whether the graph is increasing or decreasing.
Concave Up
Slopes/rates of change are ALWAYS increasing
Think: cup ∪.
The graph can actually be decreasing while concave up—the important thing is that its slopes are becoming more positive/less negative.
Concave Down
Slopes/rates of change are ALWAYS decreasing
Think: cap ∩.
Again, a graph can be increasing while concave down. Increasing/decreasing and concavity describe different things.
Point of Inflection
A point of inflection is a point where the graph's concavity changes:
concave up ↔ concave down
Straight lines have no concavity.
What to Know for 1.1
The core idea is:
Input changes → Output responds
You should be able to identify the independent/dependent variables, determine increasing/decreasing behavior from graphs/tables/equations, identify zeros and the y-intercept, distinguish concave up/down, identify inflection points, and interpret how two quantities vary in tandem.
1.2 Rates of Change
AP Precalculus 1.2 — Rates of Change
Average Rate of Change
Average rate of change measures how much the output changes compared to how much the input changes.
Average Rate of Change = change in output values / change in input values
Formula
Δy / Δx = (y2 - y1) / (x2 - x1) = slope
In function notation:
Average Rate of Change = [f(x2) - f(x1)] / (x2 - x1)
The output values are always in the numerator, and the input values are always in the denominator.
Average Rate of Change Between Two Points
If you are given two points:
(x1, y1) and (x2, y2)
Use:
(y2 - y1) / (x2 - x1)
This is the same formula used to calculate slope.
Important
Always subtract in the same order:
y2 - y1
—————
x2 - x1
If you reverse the order in the numerator, you must also reverse the order in the denominator.
Average Rate of Change From a Table
Instead of a graph or equation, you may be given a table of values.
To calculate the average rate of change on an interval:
Find the two input values that represent the interval.
Find their corresponding output values.
Calculate the change in output.
Calculate the change in input.
Divide:
change in output / change in input
Be careful that the output values are always in the numerator.
Understanding "Per"
The word per is a key indicator that you are dealing with a rate of change.
Examples:
Miles per gallon
Students per classroom
Online gamers per server
"Per" also helps identify the dependent and independent variables.
Dependent variable per Independent variable
The dependent variable is always listed first.
For example:
meters per second
meters = dependent/output variable
seconds = independent/input variable
Rate of Change at a Point
The rate of change at a point tells us about the rate at which the output values would change if the input values were to change at that point.
Unlike average rate of change, we are interested in what is happening at one specific point.
We can approximate the rate of change at a point by using average rates of change over small intervals that contain the point.
Main Idea
Smaller interval around the point → better approximation of the rate of change at that point
This represents the slope of the graph at that specific point.
Approximating Rate of Change at a Point
To estimate the rate of change at x = a:
Choose two x-values very close to x = a.
Ideally, choose one value slightly below a and one slightly above a.
Find their corresponding output values.
Calculate the average rate of change:
[f(x2) - f(x1)] / (x2 - x1)
Use that result as an approximation of the rate of change at x = a.
The closer the two x-values are to the point, the better the approximation generally becomes.
Positive Rate of Change
A positive rate of change indicates that as one quantity increases or decreases, the other quantity changes in the same direction.
In other words:
input increases → output increases
or
input decreases → output decreases
Graphically
A positive rate of change means the graph generally moves:
upward from left to right
Negative Rate of Change
A negative rate of change indicates that as one quantity increases, the other decreases.
The quantities change in opposite directions.
In other words:
input increases → output decreases
or
input decreases → output increases
Graphically
A negative rate of change means the graph generally moves:
downward from left to right
Rate of Change and Increasing/Decreasing Functions
The sign of the rate of change can tell you whether a function is increasing or decreasing over an interval.
Positive Rate of Change
Positive rate of change → function is increasing
Negative Rate of Change
Negative rate of change → function is decreasing
Zero Rate of Change
Zero rate of change → no overall change in output over the interval
If the average rate of change between two points is 0:
f(x2) - f(x1) = 0
Therefore:
f(x2) = f(x1)
The two points have the same output/y-value.
Units of Rate of Change
A rate of change should include units whenever the variables represent real quantities.
The units follow:
output units / input units
For example, if distance is measured in meters and time is measured in seconds:
meters per second
If the rate of change is:
-3 meters per second
this means the distance represented by the function is decreasing by an average of 3 meters for every 1-second increase in time.
What to Know for 1.2
The core idea is:
Rate of Change = Change in Output / Change in Input
You should be able to:
Calculate average rate of change using two points.
Calculate average rate of change from a table, graph, or equation.
Use the formula (y2 - y1) / (x2 - x1).
Recognize that average rate of change is the slope between two points.
Identify the units of a rate of change.
Understand that dependent variable per independent variable determines the units.
Approximate the rate of change at a specific point using small intervals.
Determine whether a rate of change is positive, negative, or zero.
Connect positive rates with increasing behavior and negative rates with decreasing behavior.
Recognize that an average rate of change of 0 means the two endpoints have the same output value.
1.3 Rates of Change in Linear and Quadratic Functions
AP Precalculus 1.3 — Rates of Change in Linear and Quadratic Functions
Average Rate of Change = Slope of a Secant Line
The average rate of change over the closed interval [a, b] is the slope of the secant line from the point (a, f(a)) to (b, f(b)).
Formula
Average Rate of Change = [f(b) - f(a)] / (b - a)
A secant line is a line that passes through two points on a graph.
So:
Average Rate of Change = Slope of the Secant Line
Average Rate of Change of Linear Functions
For a linear function, the average rate of change is constant.
Regardless of the input-value interval length, the average rate of change stays the same.
For a linear function:
f(x) = mx + b
The average rate of change on any interval is:
m
where m is the slope.
Main Idea
Linear function → constant average rate of change
For example, if:
f(x) = 2x + 3
then the average rate of change is always:
2
It does not matter which interval you choose.
Rate of Change of the Average Rates of Change
For a linear function, the average rates of change are always the same.
Therefore, the rate of change of the average rates of change of a linear function is:
0
Why?
If the average rates of change are:
2, 2, 2, 2, 2
then their changes are:
0, 0, 0, 0
So:
Linear function → constant average rate of change → rate of change of average rates of change = 0
Average Rate of Change of Quadratic Functions
For a quadratic function, the average rate of change does not stay the same.
For example:
f(x) = x^2
As x increases across equal-length intervals, the average rates of change also change.
Main Idea
Quadratic function → changing average rate of change
However, there is a pattern in how those average rates of change change.
Consecutive Equal-Length Intervals
When working with a quadratic function, compare the average rates of change over consecutive equal-length input-value intervals.
For example:
[1, 2]
[2, 3]
[3, 4]
[4, 5]
Each interval has the same length:
1
For consecutive equal-length input-value intervals, the rate of change of the average rates of change of a quadratic function is constant.
Example Pattern
Suppose the average rates of change are:
1, 3, 5, 7, 9
Find the change between consecutive rates:
3 - 1 = 2
5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
Therefore:
Rate of change of average rates of change = 2
The important part is that this value is constant.
Linear vs. Quadratic Functions
Linear Function
Average rate of change = constant
Therefore:
Rate of change of average rates of change = 0
Quadratic Function
Average rate of change = changing
But over consecutive equal-length intervals:
Rate of change of average rates of change = constant
This gives you a way to distinguish linear and quadratic functions from a table.
Recognizing a Linear Function From a Table
For equally spaced x-values, calculate the changes in the output values.
If the output values change by the same amount each time, the function is linear.
Example:
x: 1, 2, 3, 4
f(x): 3, 5, 7, 9
Change: +2, +2, +2
The average rate of change is constant, so the function is linear.
Recognizing a Quadratic Function From a Table
For equally spaced x-values, first find the changes in the output values.
These are the first differences.
Then find the changes between the first differences.
These are the second differences.
Example:
x: 1, 2, 3, 4, 5
f(x): 1, 4, 9, 16, 25
First differences: +3, +5, +7, +9
Second differences: +2, +2, +2
The first differences are not constant, but the second differences are constant.
Therefore, the function is quadratic.
Main Idea
Linear → constant first differences
Quadratic → constant second differences
This assumes the x-values are equally spaced.
Quadratic Functions and Concavity
The changing average rates of change can also tell you whether a quadratic function is concave up or concave down.
Concave Up
If the average rates of change are increasing, the graph is:
Concave up ∪
Example:
-4, -2, 0, 2, 4
The rates are increasing, so the graph is concave up.
The function does not have to be increasing to be concave up.
What matters is:
Average rates of change are increasing.
Concave Down
If the average rates of change are decreasing, the graph is:
Concave down ∩
Example:
5, 3, 1, -1, -3
The rates are decreasing, so the graph is concave down.
The function does not have to be decreasing to be concave down.
What matters is:
Average rates of change are decreasing.
Determining Concavity From a Table
If the x-values are equally spaced:
Step 1
Find the changes between consecutive output values.
These represent the average rates of change over equal-length intervals.
Step 2
Compare those changes.
If they are increasing → concave up
If they are decreasing → concave down
Example
x: 15, 16, 17, 18, 19
g(x): 18, 20, 20, 18, 14
Find the changes:
20 - 18 = 2
20 - 20 = 0
18 - 20 = -2
14 - 18 = -4
So the average rates of change are:
2, 0, -2, -4
They are decreasing.
Therefore:
The function is concave down.
Connection Between Rate of Change and Concavity
Remember that increasing/decreasing and concavity describe different things.
Increasing/Decreasing
Looks at whether the function's output values are increasing or decreasing.
Positive rate of change → increasing
Negative rate of change → decreasing
Concavity
Looks at whether the function's rates of change are increasing or decreasing.
Rates of change increasing → concave up
Rates of change decreasing → concave down
So a function can be:
decreasing AND concave up
or:
increasing AND concave down
What to Know for 1.3
The core ideas are:
Average Rate of Change = Slope of a Secant Line
Linear → constant average rate of change
Quadratic → changing average rate of change, but constant change in those rates over consecutive equal-length intervals
You should be able to:
Calculate average rate of change using [f(b) - f(a)] / (b - a).
Recognize average rate of change as the slope of a secant line.
Know that a linear function has a constant average rate of change.
Know that the rate of change of the average rates of change of a linear function is 0.
Know that a quadratic function's average rate of change does not stay the same.
Know that for consecutive equal-length intervals, the rate of change of the average rates of change of a quadratic function is constant.
Recognize constant first differences as linear behavior.
Recognize constant second differences as quadratic behavior.
Determine concavity by examining how the rates of change change.
Recognize increasing rates of change → concave up.
Recognize decreasing rates of change → concave down.