True or False? For a linear function, the rate at which the average rates of change are themselves changing is zero.

True. Because those average rates of change never differ from one interval to the next, there is nothing for them to change by. This second-order property is what separates a linear function from every other function type, where the rate of change itself changes.

Define local maximum .

A function has a local maximum at a point where it switches from increasing to decreasing . The output value there is greater than the output values at all nearby points, so on the graph it is the top of a hill.

An even-degree polynomial has a negative leading coefficient. Does it have a global maximum or a global minimum?

It has a global maximum . A negative leading coefficient leaves the output values bounded above but not below, so there is a highest value and no lowest one; a positive leading coefficient gives a global minimum instead.

What is the difference between a local extremum and a point of inflection?

A local extremum is where the function switches between increasing and decreasing. A point of inflection is where the rate of change of the function switches between increasing and decreasing, which is where the concavity of the graph changes.

True or False? At a point of inflection, a function must switch between increasing and decreasing.

False. A function can be increasing on both sides of a point of inflection, or decreasing on both sides. What changes there is how quickly it is increasing or decreasing, not whether it is.

A cubic polynomial has one local maximum and one local minimum. Where does its point of inflection lie?

It lies between them. The graph has to change concavity somewhere between a hill and a valley, which is why points of inflection on higher-degree polynomials often sit between local maxima and minima.

True or False? Successive differences reveal the degree of a polynomial whatever the spacing of the input values.

False. The method only works when the input values are equally spaced . With unevenly spaced inputs the differences are no longer comparing like with like, and the pattern breaks down.

What two things determine the end behavior of a polynomial?

The degree , specifically whether it is odd or even, and the sign of the leading coefficient . Nothing else about the polynomial makes any difference to its end behavior.

True or False? For an odd-degree polynomial the two ends behave in opposite ways, and for an even-degree polynomial they behave the same way.

True. Odd degree sends the two ends in opposite directions, while even degree sends them the same way. The sign of the leading coefficient then decides which way round, since a negative coefficient flips both ends compared with a positive one.

AP®MathsCollege BoardPrecalculusFlashcardsPolynomial & Rational FunctionsLinear, Quadratic & Polynomial Functions

Linear, Quadratic & Polynomial Functions (College Board AP® Precalculus): Flashcards

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FrontLinear Functions
The linear function $f$ is given by $f (x) = b + m x$ . Fill in the two gaps.
The constant $m$ is the $______$ of the line, and $b$ is the $______$ value, which is the output when $x = 0$ .

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Cards in this collection (70)

The linear function $f$ is given by $f (x) = b + m x$ . Fill in the two gaps.
The constant $m$ is the $______$ of the line, and $b$ is the $______$ value, which is the output when $x = 0$ .
The completed sentence is:
The constant $m$ is the slope of the line, and $b$ is the initial value, which is the output when $x = 0$ .
Note that this course writes the initial value first, as $f (x) = b + m x$ .
What is the average rate of change of a linear function over any interval?
It is constant, and always equal to the slope $m$ .
It does not matter which two points are chosen or how wide the interval is, and this is the defining property of a linear function.
True or False?
For a linear function, the rate at which the average rates of change are themselves changing is zero.
True.
Because those average rates of change never differ from one interval to the next, there is nothing for them to change by.
This second-order property is what separates a linear function from every other function type, where the rate of change itself changes.
In a table of values with equally spaced inputs, what shows that the function could be linear?
The differences between consecutive output values are all the same.
Dividing that constant difference by the spacing of the inputs gives the slope, so outputs rising by $6$ for every $2$ the input rises means a slope of $3$ .
What does the initial value of a linear function tell you about its graph?
It gives the $y$ -intercept, since it is the output value when $x = 0$ .
The graph of $f (x) = 3 + 2 x$ therefore crosses the $y$ -axis at $(0, 3)$ .
True or False?
A function whose average rate of change is the same over every interval must be increasing.
False.
A constant average rate of change only says the function changes at a steady rate, and that rate can be negative or zero.
A linear function with a negative slope decreases at a constant rate, and one with slope $0$ does not change at all.
A table has equally spaced inputs spaced $1$ apart, and the output values fall by $6$ each time. What type of function is this, and why?
It is best modeled by a linear function, because the rate of change over consecutive equal-length input-value intervals is constant.
Here that constant rate of change is $- 6$ , which is the slope of the linear model.
Complete the standard form of a quadratic function $f$ , including the condition on the leading coefficient.
$f (x) = a x^{2} + b x + c, where a______0$
The completed form is:
$f (x) = a x^{2} + b x + c, where a \neq 0$
If $a$ were $0$ the $x^{2}$ term would disappear and the function would be linear, not quadratic.
How does the sign of $a$ affect the graph of $f (x) = a x^{2} + b x + c$ ?
The graph is a parabola, and it opens upward into a $\cup$ shape when $a > 0$ .
When $a < 0$ it opens downward into a $\cap$ shape.
How do the average rates of change of a quadratic function behave over consecutive equal-length intervals?
They are not constant, but they change by the same amount each time, so they can themselves be described by a linear function.
For $f (x) = x^{2} + 1$ at the inputs $0, 1, 2, 3, 4$ they are $1, 3, 5, 7$ , increasing by $2$ each time.
True or False?
The average rate of change of a quadratic function is the same over every interval of the same length.
False.
That property belongs to linear functions only.
For a quadratic, the average rate of change depends on where the interval sits along the $x$ -axis, not just on how long it is.
When the inputs in a table are spaced $1$ apart, what is the average rate of change over each interval?
It is simply the difference between consecutive output values.
The denominator of $\frac{f (x_{n + 1}) - f (x_{n})}{x_{n + 1} - x_{n}}$ is then $1$ , so the fraction reduces to its numerator.
A table has equally spaced inputs, and the differences of the output values are $- 23, - 18, - 13, - 8$ . What type of function is this, and why?
It is best modeled by a quadratic function.
Those differences are not constant, but their own differences are all $5$ , so the average rates of change are changing at a constant rate.
Define the degree of a polynomial.
The degree of a polynomial is the highest power of $x$ that appears in the expression.
For $p (x) = 3 x^{4} - 2 x^{2} + x - 7$ the degree is $4$ .
For $p (x) = 3 x^{4} - 2 x^{2} + x - 7$ , what are the leading term and the leading coefficient?
The leading term is $3 x^{4}$ , which is the term with the highest power of $x$ .
The leading coefficient is $3$ , the coefficient of that leading term.
True or False?
The leading term of a polynomial is always the first term written in the expression.
False.
A polynomial does not have to be written in order of decreasing degree, so the leading term can appear anywhere.
In $p (x) = 1 - 3 x + 5 x^{3}$ the leading term is $5 x^{3}$ and the degree is $3$ .
What must every term of a polynomial function look like?
Each term must be a constant multiplied by a non-negative integer power of $x$ , with the terms added or subtracted.
The constant may be $1$ , as in $x^{3} = 1 x^{3}$ .
Fill in the two gaps about polynomials of low degree.
A constant function such as $f (x) = 5$ is a polynomial of degree $______$ and a cubic function is a polynomial of degree $______$ instead.
The completed sentence is:
A constant function such as $f (x) = 5$ is a polynomial of degree 0 and a cubic function is a polynomial of degree 3 instead.
A linear function has degree $1$ and a quadratic has degree $2$ , so all these familiar types are special cases of polynomials.
Why are $\frac{1}{x}$ , $\sqrt{x}$ and $2^{x}$ not polynomial functions?
The first two involve powers of $x$ that are not non-negative integers, since $\frac{1}{x}$ is $x^{- 1}$ and $\sqrt{x}$ is $x^{1 / 2}$ .
In $2^{x}$ the variable sits in the exponent, which makes it an exponential function instead.
Why are the degree and leading coefficient of a polynomial worth identifying?
Together they determine much of the polynomial's behavior, in particular its end behavior and its general shape.
Both can be read off as soon as you have found the term with the highest power of $x$ .
Define local maximum.
A function has a local maximum at a point where it switches from increasing to decreasing.
The output value there is greater than the output values at all nearby points, so on the graph it is the top of a hill.
Fill in the two gaps distinguishing the two kinds of maximum.
A maximum is called $______$ when its output value is greater than the output values at all nearby points, and $______$ when it is greater than every other output value of the function.
The completed sentence is:
A maximum is called local when its output value is greater than the output values at all nearby points, and global when it is greater than every other output value of the function.
A global maximum is therefore always a local maximum, but a local maximum need not be global.
True or False?
Every polynomial function has either a global maximum or a global minimum.
False.
An odd-degree polynomial such as a cubic runs off to $\infty$ in one direction and to $- \infty$ in the other, so its output values are bounded neither above nor below.
Even-degree polynomials do always have one or the other.
An even-degree polynomial has a negative leading coefficient. Does it have a global maximum or a global minimum?
It has a global maximum.
A negative leading coefficient leaves the output values bounded above but not below, so there is a highest value and no lowest one; a positive leading coefficient gives a global minimum instead.
A polynomial has real zeros at $x = - 2$ and $x = 3$ . What must lie somewhere between them?
There must be at least one local maximum or local minimum for some value of $x$ between $- 2$ and $3$ .
If the graph meets the $x$ -axis at two different points then it has to turn around at least once in between.
What is the difference between a local extremum and a point of inflection?
A local extremum is where the function switches between increasing and decreasing.
A point of inflection is where the rate of change of the function switches between increasing and decreasing, which is where the concavity of the graph changes.
True or False?
At a point of inflection, a function must switch between increasing and decreasing.
False.
A function can be increasing on both sides of a point of inflection, or decreasing on both sides.
What changes there is how quickly it is increasing or decreasing, not whether it is.
A cubic polynomial has one local maximum and one local minimum. Where does its point of inflection lie?
It lies between them.
The graph has to change concavity somewhere between a hill and a valley, which is why points of inflection on higher-degree polynomials often sit between local maxima and minima.
A polynomial is restricted to the closed interval $[a, b]$ . What happens at $x = a$ and $x = b$ ?
The output values at those included endpoints are local extrema.
Restricting the domain to a closed interval therefore creates extrema that the unrestricted polynomial does not have.
What does it mean if $(x - a)$ is a factor of a polynomial $p$ ?
It means $a$ is a zero of $p$ , so $p (a) = 0$ .
The converse holds too: if $p (a) = 0$ then $(x - a)$ must be a factor of $p$ .
Define the multiplicity of a zero.
The multiplicity of a zero is the number of times its linear factor is repeated in the factorisation of the polynomial.
In $p (x) = {(x - 1)}^{2} {(x + 4)}^{3} (x + 7)$ the zero $x = 1$ has multiplicity $2$ and $x = - 4$ has multiplicity $3$ .
How does the multiplicity of a real zero affect the graph at that $x$ -intercept?
Odd multiplicity means the graph crosses the $x$ -axis, so the output values change sign there.
Even multiplicity means the graph is tangent to the axis, touching it and turning back, with the output values keeping the same sign on both sides.
True or False?
A zero of multiplicity $3$ and a zero of multiplicity $1$ look exactly the same on the graph.
False.
Both multiplicities are odd, so the graph crosses the axis at each, but the shape differs.
At multiplicity $1$ the graph cuts straight across, while at multiplicity $3$ or more it levels out as it crosses.
Fill in the two gaps about how many zeros a polynomial has.
A polynomial function of degree $n$ has exactly $______$ complex zeros, provided they are counted with their $______$ included.
The completed sentence is:
A polynomial function of degree $n$ has exactly n complex zeros, provided they are counted with their multiplicities included.
Here a complex zero means one that is either real or non-real, so a degree $5$ polynomial always has exactly five.
$2 + 3 i$ is a zero of a polynomial with real coefficients. What else must be a zero?
The conjugate $2 - 3 i$ must also be a zero, because non-real zeros always occur in conjugate pairs.
This means non-real zeros always account for an even number of a polynomial's zeros.
True or False?
A polynomial can have zeros that never appear anywhere on its graph.
True.
Only real zeros give $x$ -intercepts, because only real numbers lie on the $x$ -axis.
A polynomial's non-real zeros are genuine zeros but do not show up on the graph at all.
How do the real zeros of a polynomial help you solve $p (x) \geq 0$ ?
They divide the $x$ -axis into intervals, and between two consecutive zeros the polynomial is entirely positive or entirely negative.
So find the zeros first, then test a single value inside each interval to determine the sign there.
How can successive differences reveal the degree of a polynomial?
Take the differences of the output values at equally spaced inputs, then the differences of those, and so on until they are constant.
The degree is the least value $n$ for which the $n$ th differences are constant.
A polynomial gives the outputs $2, 5, 14, 35, 74$ at equally spaced inputs. The 1st differences are $3, 9, 21, 39$ and the 2nd differences are $6, 12, 18$ . Fill in the two gaps.
The 3rd differences are both $______$ which is constant, so the polynomial has degree $______$ overall.
The completed sentence is:
The 3rd differences are both 6 which is constant, so the polynomial has degree 3 overall.
A polynomial of degree $3$ is a cubic.
True or False?
Successive differences reveal the degree of a polynomial whatever the spacing of the input values.
False.
The method only works when the input values are equally spaced.
With unevenly spaced inputs the differences are no longer comparing like with like, and the pattern breaks down.
Fill in the two gaps in the algebraic definitions of even and odd functions.
A function is even when $f (- x) =______$ for all values of $x$ , and odd when $f (- x) =______$ instead.
The completed sentence is:
A function is even when $f (- x) = f (x)$ for all values of $x$ , and odd when $f (- x) = - f (x)$ instead.
A single minus sign is the whole difference between the two definitions.
What symmetry does the graph of an even function have?
It is symmetric over the line $x = 0$ , so it is unchanged by a reflection in the $y$ -axis.
If the point $(x, y)$ lies on the graph then so does $(- x, y)$ .
What symmetry does the graph of an odd function have?
It is symmetric about the point $(0, 0)$ , so it is unchanged by a rotation of $180 °$ about the origin.
If the point $(x, y)$ lies on the graph then so does $(- x, - y)$ .
When is $p (x) = a_{n} x^{n}$ an even function?
When $n$ is even and $a_{n} \neq 0$ , which is why $x^{2}$ , $7 x^{6}$ and $2 x^{14}$ are all even.
It is odd when $n$ is odd instead, so $x$ , $4 x^{5}$ and $3 x^{17}$ are all odd functions.
True or False?
Adding together even functions always gives another even function.
True.
Linear combinations of even functions are themselves even, so $x^{2} + 7 x^{6} - 2 x^{14}$ is an even function.
The same holds for odd functions, which is why $x - 4 x^{5} + 3 x^{17}$ is odd.
True or False?
Only polynomial functions can be even or odd.
False.
The functions $\cos x$ and $\sec x$ are both even, since $\cos (- x) = \cos x$ .
The functions $\sin x$ , $\csc x$ , $\tan x$ and $\cot x$ are all odd.
For an even function, what corresponds to a local maximum at $(x, f (x))$ ?
There is also a local maximum at $(- x, f (- x))$ , and since $f (- x) = f (x)$ it sits at exactly the same height.
A local minimum likewise corresponds to another local minimum.
For an odd function $g$ , what corresponds to a local minimum at $(x, g (x))$ ?
There is a local maximum at $(- x, g (- x))$ , because a half turn about the origin sends a valley to a hill.
Since $g (- x) = - g (x)$ , a local minimum of $5$ at $x = 7$ gives a local maximum of $- 5$ at $x = - 7$ .
Define the end behavior of a function.
The end behavior describes what happens to a function's output values as its input values increase without bound or decrease without bound.
For a nonconstant polynomial those output values always go to $\infty$ or to $- \infty$ in each direction.
What four parts make up a complete limit statement describing end behavior?
The $\lim$ symbol, the direction such as $x \to \infty$ , the function such as $p (x)$ , and the result, $\infty$ or $- \infty$ .
A full description of a polynomial's end behavior needs two such statements, one for each direction.
Complete the limit statement meaning: as $x$ increases without bound, $p (x)$ decreases without bound. Fill in the direction underneath the limit and the value on the right.
$\lim_{x \to______} p (x) =______$
The completed statement is:
$\lim_{x \to \infty} p (x) = - \infty$
A polynomial never levels off to a horizontal asymptote, so its output values always increase or decrease without limit.
What two things determine the end behavior of a polynomial?
The degree, specifically whether it is odd or even, and the sign of the leading coefficient.
Nothing else about the polynomial makes any difference to its end behavior.
An odd-degree polynomial has a positive leading coefficient. Describe its end behavior.
The left end goes down and the right end goes up, just as for $x^{3}$ .
In limit notation, $\lim_{x \to - \infty} p (x) = - \infty$ and $\lim_{x \to \infty} p (x) = \infty$ .
True or False?
For an odd-degree polynomial the two ends behave in opposite ways, and for an even-degree polynomial they behave the same way.
True.
Odd degree sends the two ends in opposite directions, while even degree sends them the same way.
The sign of the leading coefficient then decides which way round, since a negative coefficient flips both ends compared with a positive one.
An even-degree polynomial has a negative leading coefficient. Describe its end behavior.
Both ends go down, just as for $- x^{2}$ .
In limit notation, $\lim_{x \to - \infty} p (x) = - \infty$ and $\lim_{x \to \infty} p (x) = - \infty$ .
Why does only the leading term matter for end behavior?
Higher powers of $x$ grow much faster than lower powers, so as the input values grow the leading term dominates all the others.
For $p (x) = 2 x^{4} - 50 x^{3} + 100 x$ the $2 x^{4}$ term is far larger in magnitude than the rest once $x$ is large.
Define binomial.
A binomial is an expression consisting of two terms added together, such as $(a + b)$ or $(x + 3)$ .
Expanding a binomial raised to a power means rewriting it as a polynomial in standard form.
How is each entry of Pascal's Triangle obtained?
Each entry is the sum of the two entries directly above it, with $1$ s running down both outside edges.
Rows are numbered from row $0$ at the top, so row $4$ is $1, 4, 6, 4, 1$ .
Row $4$ of Pascal's Triangle is $1, 4, 6, 4, 1$ . Fill in the two gaps in row $5$ .
Row $5$ begins $1$ , then $5$ , then $______$ then $______$ then $5$ , then $1$ at the end.
The completed row is:
Row $5$ begins $1$ , then $5$ , then 10 then 10 then $5$ , then $1$ at the end.
Each came from adding the two entries above it, giving $4 + 6 = 10$ and $6 + 4 = 10$ .
Which row of Pascal's Triangle gives the coefficients for expanding ${(a + b)}^{n}$ ?
Row $n$ , counting the single $1$ at the top as row $0$ .
That row holds $n + 1$ entries, which matches the $n + 1$ terms in the expansion.
In the expansion of ${(a + b)}^{n}$ , how do the powers of $a$ and $b$ change across the terms?
The power of $a$ decreases from $n$ down to $0$ , while the power of $b$ increases from $0$ up to $n$ .
Every term therefore has powers that add up to $n$ .
True or False?
To expand ${(x - 4)}^{4}$ you treat the subtraction as adding $- 4$ .
True.
Set $b = - 4$ and apply the usual expansion, taking care with the powers of a negative number.
A negative raised to an odd power stays negative, while a negative raised to an even power becomes positive.
Expand ${(x + 2)}^{3}$ using Pascal's Triangle.
Row $3$ gives the coefficients $1, 3, 3, 1$ , and the terms are $x^{3}$ , $x^{2} \cdot 2$ , $x \cdot 2^{2}$ and $2^{3}$ .
The expansion is:
${(x + 2)}^{3} = x^{3} + 6 x^{2} + 12 x + 8$
What information is easiest to read from a polynomial in standard form?
The degree, the leading term and the leading coefficient, and therefore the end behavior.
The constant term $a_{0}$ also gives the $y$ -intercept directly, at the point $(0, a_{0})$ .
What information is easiest to read from a polynomial in factored form?
The real zeros, the multiplicity of each one, and therefore the $x$ -intercepts of the graph.
From $p (x) = - 2 (x - 1) (x + 3) (x - 5)$ the $x$ -intercepts are $(1, 0)$ , $(- 3, 0)$ and $(5, 0)$ .
How do you convert a polynomial from factored form to standard form?
Expand the factors and collect like terms.
For example, $(x - 2) (x + 5) = x^{2} + 5 x - 2 x - 10 = x^{2} + 3 x - 10$ .
Complete the expansion converting this polynomial into standard form.
$(x + 4) (x - 4) (x + 2) = x^{3} +______x^{2} -______x - 32$
The completed expansion is:
$(x + 4) (x - 4) (x + 2) = x^{3} + 2 x^{2} - 16 x - 32$
Multiplying the first two factors gives $x^{2} - 16$ , which makes the rest of the expansion much shorter.
How do you factor $x^{3} + 2 x^{2} - 15 x$ without technology?
First take out the common factor $x$ , then factor the quadratic that is left.
This gives $x (x^{2} + 2 x - 15) = x (x + 5) (x - 3)$ .
True or False?
A polynomial written in factored form is always a product of linear factors only.
False.
There may also be an irreducible quadratic factor, which is what happens when the polynomial has non-real zeros.
For example, $p (x) = 3 (x - 2) (x^{2} + 2 x + 5)$ is in factored form.
You have $p (x) = - 2 (x - 1) (x + 3) (x - 5)$ and need its degree. Must you expand it fully?
No, multiply only the leading term of each factor, giving $- 2 \cdot x \cdot x \cdot x = - 2 x^{3}$ .
So the degree is $3$ and the leading coefficient is $- 2$ , with nothing else expanded.

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