Unit 1 · Topic 1.4

Polynomial Functions and Rates of Change

A polynomial function has a degree and a leading term.

Practice 2: Multiple RepresentationsPractice 3: Communication and Reasoning

Part 1 · Hook

Why this matters

A greenhouse warms all morning, peaks in the late afternoon and cools at night. A polynomial function can follow that day: its graph turns at the warmest moment, and it bends where the warming starts to slow. This topic names those turning and bending points and finds them on graphs, tables and a calculator.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which is $x^3 - 9x$ factored completely?

$x(x - 3)(x + 3)$
$x(x^2 - 9)$
$(x - 3)(x + 3)$
$x(x - 9)$

Show the answer

Common factor $x$ , then a difference of squares.

Correct: $x(x - 3)(x + 3)$ :
$x(x^2 - 9)$ :
$(x - 3)(x + 3)$ :
$x(x - 9)$ :

2. The average rates of change of $f$ over consecutive equal intervals are $6, 4, 2$ . The graph of $f$ is

concave down
concave up
decreasing
linear

Show the answer

Decreasing rates mean concave down.

Correct: concave down:
concave up:
decreasing:
linear:

Part 4 · See it

See it first

The graph of f(x) = x³ − 6x² + 9x for −0.4 ≤ x ≤ 4.4. It increases to a local maximum at (1, 4), decreases to a local minimum at (3, 0), then increases. Its point of inflection is (2, 2), where it changes from concave down to concave up. — $f(x) = x^3 - 6x^2 + 9x$ has a local maximum $(1, 4)$ , a local minimum $(3, 0)$ and a point of inflection $(2, 2)$ , where it changes from concave down to concave up. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

A polynomial function switches from increasing to decreasing at $x = a$ it has a local maximum at $a$ (decreasing to increasing gives a local minimum)
The domain is limited to a closed intervalan included endpoint can also be a local extremum, so check the endpoints
You compare every local extremum value and every endpoint valuethe greatest is the global maximum and the least is the global minimum
The rate of change switches from increasing to decreasing, or backthe graph changes concavity: a point of inflection

Part 6 · Key ideas

Key ideas

A polynomial function $p(x)$ $= a_n x^n + \cdots + a_1 x + a_0$ has degree $n$ and leading term $a_n x^n$ .
Local maximum: an output at least as large as at nearby inputs; local minimum: at most as large. Global: over the whole domain.
Between two distinct real zeros there is at least one local extremum. An even-degree polynomial has a global maximum or minimum.
A point of inflection is where the graph changes concavity: the rate of change switches from increasing to decreasing, or back.

Part 7 · Misconception

A common mistake

The wrong idea: The global maximum is the largest local maximum.

What actually happens: Check the endpoints too. On $[-3, 2.5]$ , $p(x) = \frac{x^4}{4} - 2x^2 + 3$ has a local maximum $3$ at $x = 0$ , but $p(-3) = 5.25$ is the global maximum.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graph of $p$ on $[-3, 2.5]$

1. The function $p(x) = \frac{x^4}{4} - 2x^2 + 3$ is defined only for $-3 \le x \le 2.5$ . At which inputs does $p$ have a local minimum?

$x = -2$ and $x = 2$
$x = 0$
$x = -3$ and $x = 0$
$x = -1$ and $x = 1$

Show the answer

A local minimum is where the output is less than or equal to the outputs at nearby inputs: where $p$ changes from decreasing to increasing, at $x = -2$ and $x = 2$ (both with $p = -1$ ).

Correct: $x = -2$ and $x = 2$ : Right: $p$ switches from decreasing to increasing at both.
$x = 0$ : At $x = 0$ , $p$ switches from increasing to decreasing: a local maximum.
$x = -3$ and $x = 0$ : These are local maxima: the output there is greater than at every nearby input.
$x = -1$ and $x = 1$ : At $x = -1$ the graph is still rising and at $x = 1$ it is still falling; it does not turn around at either, so neither is a local minimum.

2. The function $p(x) = \frac{x^4}{4} - 2x^2 + 3$ is defined only for $-3 \le x \le 2.5$ . What is the global maximum value of $p$ on its domain?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Compare the local maximum values and the endpoint values: $p(0) = 3$ , $p(-3)$ $= \frac{81}{4} - 18 + 3$ $= 5.25$ and $p(2.5) \approx 0.27$ . The greatest is $5.25$ , at the included endpoint $x = -3$ .

Answer: 5.25

3. The function $p(x) = \frac{x^4}{4} - 2x^2 + 3$ is defined only for $-3 \le x \le 2.5$ . How many points of inflection does the graph of $p$ have?

One
Two
Three
Four

Show the answer

A point of inflection is where the graph changes concavity. $p$ is concave up near $x = -2$ , concave down near $x = 0$ and concave up near $x = 2$ : two switches, at about $x = -1.15$ and $x = 1.15$ .

One: Look at both sides: the graph changes from concave up to concave down left of $x = 0$ and back again right of it.
Correct: Two: Right: concave up, then concave down around $x = 0$ , then concave up again; it switches twice.
Three: The turning points $-2$ , $0$ and $2$ are extrema, not inflection points.
Four: Count where the bending changes, not where the direction changes.

Graph

Temperature in a greenhouse

4. The temperature in a greenhouse is $T(h)$ $= -0.015h^3 + 0.4h^2 + 52$ degrees Fahrenheit, $h$ hours after midnight, for $0 \le h \le 24$ . What is the greatest temperature on $[0, 24]$ ? Give it to three decimal places.

Calculator allowed. Give the answer correct to three decimal places, in degrees Fahrenheit.

Show the answer

The maximum feature gives $T \approx 94.140$ degrees at $h \approx 17.778$ . Evaluating at a rounded $h = 18$ gives $94.12$ , which is not accurate to three decimals.

Answer: 94.140 degrees Fahrenheit

5. The temperature in a greenhouse is $T(h)$ $= -0.015h^3 + 0.4h^2 + 52$ degrees Fahrenheit, $h$ hours after midnight, for $0 \le h \le 24$ . Which describes $T$ on the interval $(12, 17)$ ?

$T$ is increasing, and the rate of change of $T$ is increasing.
$T$ is decreasing, and the rate of change of $T$ is decreasing.
$T$ is increasing, and the rate of change of $T$ is decreasing.
$T$ is decreasing, and the rate of change of $T$ is increasing.

Show the answer

On $(12, 17)$ the graph rises, so $T$ is increasing. The hourly changes shrink (about $2.97$ from $12$ to $13$ , about $0.95$ from $16$ to $17$ ), so the rate of change of $T$ is decreasing: $T$ is concave down there.

$T$ is increasing, and the rate of change of $T$ is increasing.: The graph bends downward on $(12, 17)$ , so the rates are falling. Compare $\frac{T(13) - T(12)}{1}$ with $\frac{T(17) - T(16)}{1}$ .
$T$ is decreasing, and the rate of change of $T$ is decreasing.: $T$ keeps rising until about $h = 17.778$ .
Correct: $T$ is increasing, and the rate of change of $T$ is decreasing.: Right: the temperature rises, but by less each hour as it nears its peak.
$T$ is decreasing, and the rate of change of $T$ is increasing.: The temperature rises on $(12, 17)$ ; it falls only after $h \approx 17.778$ .

6. A polynomial function $f$ has zeros at $x = 1$ and $x = 4$ and no zeros between them. Which must be true?

$f$ is increasing on $[1, 4]$ .
$f$ has a local maximum or a local minimum between $1$ and $4$ .
$f(2.5) = 0$
$f$ has a point of inflection at $x = 2.5$ .

Show the answer

Between two distinct real zeros, a polynomial function's graph must turn around to come back to the $x$ -axis, so it has at least one local extremum there.

$f$ is increasing on $[1, 4]$ .: $f$ starts and ends at $0$ on $[1, 4]$ , so it cannot rise the whole way.
Correct: $f$ has a local maximum or a local minimum between $1$ and $4$ .: Right: $f$ leaves $0$ and comes back to $0$ , so it turns around in between.
$f(2.5) = 0$ : There are no zeros between $1$ and $4$ .
$f$ has a point of inflection at $x = 2.5$ .: Nothing forces a change in concavity at the midpoint.

Part 9 · Summary

Summary

A polynomial function has a degree and a leading term. Where it switches between increasing and decreasing it has a local maximum or minimum; on a closed interval the endpoints count too, and the greatest and least of all these values are the global maximum and minimum. Where its rate of change switches between increasing and decreasing, the graph has a point of inflection.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graph of pp on [−3,2.5][-3, 2.5]

Temperature in a greenhouse

Summary

What comes next

Connections

The graph of $p$ on $[-3, 2.5]$