A function is a mathematical relation that maps a set of input values to a set of output values, with each input mapped to exactly one output. If a single input value can produce more than one output value, the relation is not a function.

State what the domain and range of a function are, and name the variable that represents each.

The domain is the set of all input values, represented by the independent variable . The range is the set of all output values, represented by the dependent variable , so called because its value depends on the value chosen for the input.

How can you tell from a graph whether it represents a function?

Use the vertical line test . If any vertical line crosses the graph at more than one point the graph does not represent a function, because that vertical line is a single input value being mapped to more than one output.

What is the rate of change of a straight-line graph equal to?

It is equal to the slope of the line. A positive slope means the graph is increasing, and a negative slope means it is decreasing.

A quantity increases at a constant rate. What shape is that section of its graph, and what does a steeper section mean?

A constant rate of change gives a straight section on the graph. The steeper that section is, the faster the quantity is changing.

Reading a graph from left to right, what does an increasing section look like?

It goes up as it moves to the right, from bottom left to top right. A decreasing section goes down as it moves to the right, from top left to bottom right.

Why is a function described as increasing on an interval rather than simply as increasing?

Most functions increase on some parts of their domain and decrease on others, so the description only means something once the interval is named. The same function can be increasing on one interval and decreasing on the next.

Define what it means for a graph to be concave up .

The graph of a function is concave up on an interval where its rate of change is increasing . That can mean a positive rate of change becoming more positive, or a negative rate of change becoming less negative.

A quantity is increasing at a decreasing rate . What does that say about its graph?

The function is increasing, so its rate of change is positive, and that rate of change is getting smaller. The graph is therefore rising and concave down .

True or False? Knowing that a graph is concave up tells you that the function is increasing.

False. Concavity describes how the rate of change is behaving, not what sign it has. A concave up graph can belong to a function that is increasing or to one that is decreasing.

Define point of inflection .

A point of inflection is a point where the graph of a function changes from concave down to concave up , or from concave up to concave down. It is where the rate of change stops increasing and starts decreasing, or the other way round.

Define the average rate of change of a function over an interval.

It is the constant rate of change that would produce the same overall change in output values as the function actually produces over that interval. It is the ratio of the change in output values to the change in input values.

On a graph, what is the rate of change at a point equal to?

It is equal to the slope of the tangent line to the graph at that point. The steeper that tangent line is, the greater the rate of change at the point.

What does a negative rate of change say about how two quantities vary together?

They move in opposite directions , so as one increases the other decreases. For example, if the rate of change of a test score with respect to time since last studying is negative, the score falls as time passes.

True or False? A positive rate of change means that if one quantity decreases, the other decreases too.

True. A positive rate of change means the two quantities move in the same direction , whichever way that is. So a decrease in the input goes with a decrease in the output, just as an increase goes with an increase.

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Functions & Rates of Change (College Board AP® Precalculus): Flashcards

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Cards in this collection (41)

Define function.
A function is a mathematical relation that maps a set of input values to a set of output values, with each input mapped to exactly one output.
If a single input value can produce more than one output value, the relation is not a function.
State what the domain and range of a function are, and name the variable that represents each.
The domain is the set of all input values, represented by the independent variable.
The range is the set of all output values, represented by the dependent variable, so called because its value depends on the value chosen for the input.
True or False?
If two different input values of a function give the same output value, the relation is not a function.
False.
Two different inputs are allowed to map to the same output, so a function with $f (2) = 7$ and $f (5) = 7$ is perfectly valid.
What is not allowed is one input mapping to two different outputs.
How can you tell from a graph whether it represents a function?
Use the vertical line test.
If any vertical line crosses the graph at more than one point the graph does not represent a function, because that vertical line is a single input value being mapped to more than one output.
The function $f$ is given by $f (x) = x^{2} - 4 x + 7$ with domain $\{1, 2, 3\}$ . Fill in the two gaps.
Both $f (1)$ and $f (3)$ are equal to $______$ so, because a repeated output value is listed only once, the range of $f$ contains exactly $______$ values in total.
The completed sentence is:
Both $f (1)$ and $f (3)$ are equal to 4 so, because a repeated output value is listed only once, the range of $f$ contains exactly 2 values in total.
The range of $f$ is $\{3, 4\}$ .
What two conditions must hold for two functions $f$ and $g$ to be equal?
They must have the same domain, and they must give the same output for every input value in that domain.
Both conditions are required, so $f (x) = x^{2}$ defined for all real numbers and $g (x) = x^{2}$ defined for $0 \leq x \leq 3$ are not equal.
True or False?
Two functions given by different-looking expressions can never be equal functions.
False.
Two functions written differently are equal provided they share the same domain and give the same output for every input in it.
For example $f (x) = {(x + 1)}^{2}$ and $g (x) = x^{2} + 2 x + 1$ , both defined for all real numbers, are equal.
For a function, what is the image of an input value and what is the preimage of an output value?
The image of an input is the single output value the function produces for it, so for $f (x) = x^{2}$ the image of $3$ is $9$ .
The preimage of an output is the set of all input values producing it, so the preimage of $9$ is $\{- 3, 3\}$ .
True or False?
A function can be represented in words alone, with no equation.
True.
A function rule can be expressed analytically as an equation, graphically as a curve, numerically as a table, or verbally in words.
The description "the output is five less than three times the input" gives the same function as $f (x) = 3 x - 5$ .
The point $(19, 28)$ lies on the graph of $f$ . What does this tell you about $f$ ?
It tells you that $f (19) = 28$ .
On the graph of $y = f (x)$ each point has an input value as its $x$ coordinate and the corresponding output value as its $y$ coordinate.
Define zero of a function.
A zero of a function is an input value whose output value is $0$ .
On a graph these are the $x$ coordinates of the points where the curve meets the $x$ -axis.
What is the rate of change of a straight-line graph equal to?
It is equal to the slope of the line.
A positive slope means the graph is increasing, and a negative slope means it is decreasing.
True or False?
At a zero of a function, the graph can touch the $x$ -axis without crossing it.
True.
A zero is any input value whose output value is $0$ , so the graph meets the $x$ -axis there whether it crosses or only touches.
Both cases count as zeros.
If you know the equation of a function, how do you find its zeros?
Set the function equal to $0$ and solve for $x$ .
For $f (x) = x^{2} - 9$ , solving $x^{2} - 9 = 0$ gives the zeros $x = - 3$ and $x = 3$ .
Complete the working to find the zeros of $f (x) = x^{2} - 2 x - 8$ .
$x^{2} - 2 x - 8 = 0 \Rightarrow (x +______) (x -______) = 0$
The completed working is:
$x^{2} - 2 x - 8 = 0 \Rightarrow (x + 2) (x - 4) = 0$
The zeros of $f$ are $x = - 2$ and $x = 4$ .
A quantity increases at a constant rate. What shape is that section of its graph, and what does a steeper section mean?
A constant rate of change gives a straight section on the graph.
The steeper that section is, the faster the quantity is changing.
On a graph of volume against time, what does a horizontal section tell you?
The volume is not changing during that time.
A horizontal section has a slope of $0$ , so the output value stays the same as the input value increases.
Define what it means for a function to be increasing on an interval.
A function $f$ is increasing on an interval if, for all $a$ and $b$ in that interval, $a < b$ implies $f (a) < f (b)$ .
In words, as the input values increase the output values always increase.
Reading a graph from left to right, what does an increasing section look like?
It goes up as it moves to the right, from bottom left to top right.
A decreasing section goes down as it moves to the right, from top left to bottom right.
True or False?
A function is increasing on an interval as soon as its output value at the right-hand end is greater than at the left-hand end.
False.
The definition requires $f (a) < f (b)$ for every pair of values $a < b$ in the interval, not just for the two endpoints.
A function that rises, dips and then rises again is not increasing on an interval containing the dip.
Complete the formal definition of a decreasing function, where $a$ and $b$ are any two values in the interval.
$a < b \Rightarrow f (a)______f (b)$
The completed definition is:
$a < b \Rightarrow f (a) > f (b)$
As the input values increase, the output values always decrease.
Why is a function described as increasing on an interval rather than simply as increasing?
Most functions increase on some parts of their domain and decrease on others, so the description only means something once the interval is named.
The same function can be increasing on one interval and decreasing on the next.
True or False?
If a function is increasing on an interval, it cannot take the same output value twice on that interval.
True.
For any two different input values $a < b$ in the interval the definition gives $f (a) < f (b)$ , so the two output values cannot be equal.
Every output value on that interval therefore comes from exactly one input value.
Define what it means for a graph to be concave up.
The graph of a function is concave up on an interval where its rate of change is increasing.
That can mean a positive rate of change becoming more positive, or a negative rate of change becoming less negative.
Fill in the two gaps in the description of a graph that is concave down.
The graph of a function is concave down on an interval where its rate of change is $______$ so a negative rate of change is becoming $______$ negative there.
The completed description is:
The graph of a function is concave down on an interval where its rate of change is decreasing so a negative rate of change is becoming more negative there.
A positive rate of change that is becoming less positive also gives a concave down graph.
A quantity is increasing at a decreasing rate. What does that say about its graph?
The function is increasing, so its rate of change is positive, and that rate of change is getting smaller.
The graph is therefore rising and concave down.
True or False?
Knowing that a graph is concave up tells you that the function is increasing.
False.
Concavity describes how the rate of change is behaving, not what sign it has.
A concave up graph can belong to a function that is increasing or to one that is decreasing.
Define point of inflection.
A point of inflection is a point where the graph of a function changes from concave down to concave up, or from concave up to concave down.
It is where the rate of change stops increasing and starts decreasing, or the other way round.
Which shape, $\cup$ or $\cap$ , corresponds to a concave down section of a graph?
The $\cap$ shape, like an upside-down cup.
A concave up section is the $\cup$ shape, which is the way round that would hold water.
Define the average rate of change of a function over an interval.
It is the constant rate of change that would produce the same overall change in output values as the function actually produces over that interval.
It is the ratio of the change in output values to the change in input values.
What does the average rate of change of $f$ over $[a, b]$ mean on the graph of $f$ ?
It is the slope of the secant line joining the points $(a, f (a))$ and $(b, f (b))$ .
A secant line is a straight line connecting two points on the graph of a function.
Complete the formula for the average rate of change of $f$ over the interval $[a, b]$ .
$Average rate of change = \frac{f (b) -______}{______- a}$
The completed formula is:
$Average rate of change = \frac{f (b) - f (a)}{b - a}$
This is the change in output divided by the change in input, sometimes written as $\frac{Δ y}{Δ x}$ .
True or False?
If the average rate of change of $f$ over an interval is positive, then $f$ is increasing at every point of that interval.
False.
The average rate of change compares only the output values at the two ends of the interval, so it describes the net change across the interval as a whole.
The function may rise and fall repeatedly inside the interval and still have a positive average rate of change.
What does the sign of an average rate of change tell you about the secant line?
A positive average rate of change gives a secant line sloping upward, so the output increases overall across the interval.
A negative one slopes downward, and an average rate of change of $0$ gives a horizontal secant, so the output values at the two ends are equal.
The average rate of change of a population over $[2, 6]$ hours is $3$ thousand per hour. What does that mean?
On average the population increases by $3$ thousand for each $1$ hour increase in time across that interval.
The units of an average rate of change are always the output units per one unit of the input, here thousands per hour.
On a graph, what is the rate of change at a point equal to?
It is equal to the slope of the tangent line to the graph at that point.
The steeper that tangent line is, the greater the rate of change at the point.
True or False?
A rate of change of $- 9$ is a faster rate of change than a rate of change of $- 4$ .
True.
How fast a quantity is changing is measured by the absolute value of the rate of change, and $| - 9 | = 9 > 4 = | - 4 |$ .
The sign tells you only the direction, and both of these describe a decreasing function.
How can you approximate the rate of change of $f$ at $x = 3$ using average rates of change?
Calculate the average rate of change over a small interval containing $x = 3$ , such as $[2.9, 3.1]$ .
The smaller the interval, the better the approximation.
To compare the rates of change at two different points, do this around each point and compare the two results.
What does a negative rate of change say about how two quantities vary together?
They move in opposite directions, so as one increases the other decreases.
For example, if the rate of change of a test score with respect to time since last studying is negative, the score falls as time passes.
Fill in the two gaps in the test for concavity using average rates of change over successive equal-length intervals.
If those average rates of change are $______$ then the graph is concave $______$ over that region.
The completed test is:
If those average rates of change are increasing then the graph is concave up over that region.
If instead the average rates of change are decreasing, the graph is concave down.
True or False?
A positive rate of change means that if one quantity decreases, the other decreases too.
True.
A positive rate of change means the two quantities move in the same direction, whichever way that is.
So a decrease in the input goes with a decrease in the output, just as an increase goes with an increase.

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