Unit 1.1-1.3

What is a function?

A relation mapping each input to exactly one output.

What is the domain of a function?

The set of input values.

What is another name for the input variable?

The independent variable, usually x.

What is the range of a function?

The set of output values.

What is another name for the output variable?

The dependent variable, usually y.

Why is y called the dependent variable?

Its value depends on the independent variable.

In h(t), what is t?

The input or independent variable.

In h(t), what is h?

The output or dependent variable.

What does it mean for two variables to vary in tandem?

Changes in one variable correspond to changes in the other according to a function rule.

What does VANG stand for?

Verbally, Analytically, Numerically, Graphically.

What is a verbal function representation?

A description of the relationship using words.

What is an analytical function representation?

A representation using an equation.

What is a numerical function representation?

A representation using a table of values.

What is a graphical function representation?

A representation using a graph.

When is a function increasing?

When output increases as input increases.

How does an increasing function move graphically?

Upward from left to right.

If a<b for an increasing function, how do f(a) and f(b) compare?

f(a)<f(b).

When is a function decreasing?

When output decreases as input increases.

How does a decreasing function move graphically?

Downward from left to right.

If a<b for a decreasing function, how do f(a) and f(b) compare?

f(a)>f(b).

How do you test increasing/decreasing behavior from a table?

Read x from smallest to largest and observe how f(x) changes.

What does consistently rising f(x) indicate?

The function is increasing.

What does consistently falling f(x) indicate?

The function is decreasing.

What is a zero of a function?

An x-value where f(x)=0.

Where does a graph intersect the x-axis?

At points where f(x)=0.

What do zeros correspond to graphically?

x-intercepts.

When does a graph intersect the y-axis?

When the input x=0.

What is the y-intercept's output value?

f(0).

What is concavity?

How a graph bends or how its rate of change changes.

What defines concave up?

The slopes or rates of change are increasing.

What shape helps you remember concave up?

A cup: ∪.

Can a decreasing function be concave up?

Yes. Its slopes must become more positive or less negative.

What defines concave down?

The slopes or rates of change are decreasing.

What shape helps you remember concave down?

A cap: ∩.

Can an increasing function be concave down?

Yes. Its rates of change can decrease while remaining positive.

Are increasing/decreasing and concavity the same?

No. One tracks outputs; the other tracks rates of change.

What is a point of inflection?

A point where the graph changes concavity.

What concavity change occurs at an inflection point?

Concave up ↔ concave down.

Do straight lines have concavity?

No.

What is the core idea of change in tandem?

Input changes → output responds.

What should you identify when analyzing variables in a function?

Which variable is independent and which is dependent.

What behaviors should you identify from graphs, tables, or equations?

Whether the function is increasing or decreasing.

What key graph features should you identify in Unit 1.1?

Zeros, the y-intercept, concavity, and inflection points.

What is average rate of change?

How much the output changes compared with how much the input changes.

What is the basic average rate of change formula?

Change in output ÷ change in input.

What is Δy/Δx equal to?

(y₂−y₁)/(x₂−x₁), which is slope.

What is AROC in function notation?

[f(x₂)−f(x₁)]/(x₂−x₁).

Which values go in the numerator of a rate of change?

Output values.

Which values go in the denominator of a rate of change?

Input values.

How do you find AROC between (x₁,y₁) and (x₂,y₂)?

(y₂−y₁)/(x₂−x₁).

What familiar calculation is AROC between two points?

Slope.

What subtraction rule must you follow when calculating AROC?

Subtract the numerator and denominator in the same order.

If you reverse subtraction in the numerator, what must you do?

Reverse subtraction in the denominator too.

How do you start finding AROC from a table?

Find the two input values that define the interval.

After finding the table's input values, what comes next?

Find their corresponding output values.

What changes do you calculate for AROC from a table?

Change in output and change in input.

How do you finish AROC from a table?

Divide change in output by change in input.

What word often signals a rate of change?

Per.

What are examples of rates using "per"?

Miles per gallon, students per classroom, gamers per server.

How are rate units ordered?

Dependent variable per independent variable.

Which variable is listed first in a "per" rate?

The dependent/output variable.

In meters per second, what is the dependent variable?

Meters.

In meters per second, what is the independent variable?

Seconds.

What does rate of change at a point describe?

How the output would change as the input changes at that specific point.

How does rate at a point differ from AROC?

It focuses on one point rather than an interval.

How can rate of change at a point be approximated?

Use average rates of change over small intervals containing the point.

What happens as the interval around a point gets smaller?

The rate-at-a-point approximation generally improves.

What does rate of change at a point represent graphically?

The slope of the graph at that point.

How do you begin approximating rate at x=a?

Choose two x-values very close to a.

Where should the two x-values ideally be relative to a?

One slightly below a and one slightly above a.

After choosing nearby x-values, what do you find?

Their corresponding output values.

What formula estimates rate at x=a?

[f(x₂)−f(x₁)]/(x₂−x₁) using nearby x-values.

What does a positive rate of change mean?

The two quantities change in the same direction.

If input increases with a positive rate, what happens to output?

Output increases.

If input decreases with a positive rate, what happens to output?

Output decreases.

How does a positive rate appear graphically?

The graph moves upward from left to right.

What does a negative rate of change mean?

The quantities change in opposite directions.

If input increases with a negative rate, what happens to output?

Output decreases.

If input decreases with a negative rate, what happens to output?

Output increases.

How does a negative rate appear graphically?

The graph moves downward from left to right.

What does a positive rate indicate about function behavior?

The function is increasing over the interval.

What does a negative rate indicate about function behavior?

The function is decreasing over the interval.

What does a zero average rate of change mean?

There is no overall change in output between the endpoints.

If AROC=0, what equation is true?

f(x₂)−f(x₁)=0.

If AROC=0, how do endpoint outputs compare?

f(x₂)=f(x₁).

If AROC=0, what do the endpoints share?

The same output or y-value.

When should rate of change include units?

When the variables represent real quantities.

What is the unit structure for rate of change?

Output units per input units.

If distance is meters and time is seconds, what are rate units?

Meters per second.

What does −3 meters per second mean?

Distance decreases by an average of 3 meters for each 1-second increase in time.

From what representations can AROC be calculated?

A table, graph, or equation.

What is AROC geometrically?

The slope between two points.

How do you determine rate-of-change units?

Dependent/output units per independent/input units.

What signs should you recognize for rates of change?

Positive, negative, and zero.

What is AROC over the closed interval [a,b]?

The slope of the secant line from (a,f(a)) to (b,f(b)).

What is the AROC formula over [a,b]?

[f(b)−f(a)]/(b−a).

What is a secant line?

A line passing through two points on a graph.

What is the relationship between AROC and a secant line?

AROC equals the slope of the secant line.

How does AROC behave for a linear function?

It is constant.

Does interval length change a linear function's AROC?

No. The AROC stays the same.