Task Sets: Compare exponential and polynomial growth - Common Core Math - HS Functions

Concepts

Which statement best describes what "eventually exceeds" means in
the principle that exponential growth eventually exceeds polynomial
growth?

The exponential function is larger than the polynomial for every
value of x.

There is some value of x beyond which the exponential function
is always larger than the polynomial.

The exponential function is larger than the polynomial for about
half of all x-values.

The exponential function grows faster only when its base is
greater than 2.

A student says: "Since $x^2$ grows faster and faster (unlike
$x$ ), $x^2$ must be an exponential function."

What is wrong with this reasoning?

Nothing is wrong — $x^2$ does grow exponentially because its
rate of change increases.

Exponential functions require a constant multiplicative ratio
between consecutive values; $x^2$ does not have a constant
ratio.

$x^2$ cannot grow faster and faster because it is a polynomial.

Exponential functions must have integer bases, so $x^2$ is not
exponential only because 2 is the exponent, not the base.

The table below shows values for $f(x) = 1000x$ and $g(x) = 2^x$ .

$x$	$f(x) = 1000x$	$g(x) = 2^x$
1	1,000	2
5	5,000	32
10	10,000	1,024
15	15,000	32,768

At which $x$ -value shown in the table does $g(x) = 2^x$ first
exceed $f(x) = 1000x$ ?

Procedures

Use the table to compare $f(x) = 100x$ and $g(x) = 3^x$ .

$x$	$f(x) = 100x$	$g(x) = 3^x$
1	100	3
2	200	9
3	300	27
4	400	81
5	500	243
6	600	729

At which $x$ -value shown in the table does $g(x) = 3^x$ first
exceed $f(x) = 100x$ ?

Consider $p(x) = x^2$ and $g(x) = 2^x$ .

Evaluate both functions at $x = 5$ .

What is $g(5) - p(5)$ ? (Enter a positive number if $g$ is larger,
negative if $p$ is larger.)

The table compares $p(x) = x^3$ and $g(x) = 2^x$ :

$x$	$p(x) = x^3$	$g(x) = 2^x$
8	512	256
9	729	512
10	1,000	1,024
11	1,331	2,048

Between which two consecutive integers does the crossover occur?

Between $x = 8$ and $x = 9$

Between $x = 9$ and $x = 10$

Between $x = 10$ and $x = 11$

There is no crossover; $x^3$ is always larger.

For $f(x) = 1000x$ and $g(x) = 2^x$ , which of the following
best explains why the exponential eventually overtakes the linear
function?

The exponential adds a constant amount each step, and that
amount eventually exceeds 1000.

The exponential multiplies by 2 each step, so its increment
at each step equals its current value and grows exponentially,
while the linear function adds only 1000 each step.

The exponential grows by a fixed percentage that is larger
than the linear function's slope.

The coefficient 1000 shrinks over time, making the linear
function grow more slowly.

HSF.LE.A.3 Tutor Intake — Exponential vs. Polynomial Growth

Concepts

Procedures