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

A

Warm-Up: Review What You Know

1.

Compare $f(x) = 100x$ and $g(x) = 2^x$ at $x = 5$ . Which is larger?

A.

$g(x) = 2^x$ , because exponential functions are always larger.

B.

$f(x) = 100x$ , because the linear function is larger at $x = 5$ ( $f(5) = 500$ vs $g(5) = 32$ ).

C.

They are equal at every point.

D.

Neither — both are undefined at $x = 5$ .

2.

Evaluate $g(x) = 2^x$ at $x = 20$ . What is $2^{20}$ ?

3.

A student claims $h(x) = x^2$ grows exponentially because it gets faster and faster. Is the student correct?

A.

Yes — $x^2$ accelerates (increases at an increasing rate), so it must be exponential.

B.

No — exponential growth requires CONSTANT RATIOS between consecutive outputs. The ratios of $x^2$ are not constant.

C.

Yes — any function that grows faster and faster is exponential.

D.

No — $x^2$ does not get faster and faster; it grows at a steady linear rate.

B

Fluency Practice

1.

At what value of $x$ does $g(x) = 2^x$ first exceed $f(x) = 1000x$ ?

Complete the table and find the crossover:

$x$	$f(x) = 1000x$	$g(x) = 2^x$
13	13,000	8,192
14	14,000	16,384
15	15,000	32,768

Enter the first integer $x$ where $g(x) > f(x)$ .

2.

The table shows values of $f(x) = 500x$ and $g(x) = 1.5^x$ :

$x$	$f(x) = 500x$	$g(x) = 1.5^x$
20	10,000	3,325
25	12,500	25,251
30	15,000	191,751

Which statement is correct?

A.

The linear function will always be larger because it has a bigger coefficient (500 vs 1.5).

B.

The crossover occurs between $x = 20$ and $x = 25$ , after which the exponential dominates.

C.

The exponential is always larger because $1.5^{25} > 500 \times 25$ .

D.

Neither function dominates — they oscillate in which is larger.

3.

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

At what value of $x$ do they cross for the final time (after which $g > p$ permanently)?
Use this table:

$x$	$p(x) = x^2$	$g(x) = 2^x$
2	4	4
3	9	8
4	16	16
5	25	32

Enter the value of $x$ at the last crossing.

4.

Can the exponential function $g(x) = 1.01^x$ eventually exceed $p(x) = x^{100}$ ?

A.

No — the base 1.01 is too close to 1 for the exponential to ever catch up.

B.

No — $x^{100}$ has such a high degree that no exponential can exceed it.

C.

Yes — for any base $b > 1$ , the exponential $b^x$ eventually exceeds every polynomial, regardless of the polynomial's degree or coefficients.

D.

Only if the base is at least 2.

5.

On a graph showing both $f(x) = x^3$ and $g(x) = 2^x$ for $x > 0$ , which region description is correct?

A.

$f(x) > g(x)$ for all $x > 0$ .

B.

$g(x) > f(x)$ for all $x > 0$ .

C.

$f(x) > g(x)$ for small $x$ values (around $x = 5$ , say), but $g(x) > f(x)$ for all sufficiently large $x$ .

D.

$g(x) > f(x)$ for all $x$ between 0 and 5, then $f(x) > g(x)$ afterwards.

C

Mixed Practice

D

Word Problems

1.

Job A pays $40,000 per year with a $2,000 annual raise (linear). Job B pays $30,000 per year with an 8% annual raise (exponential).

In what year (after starting, i.e., year 0 is the starting salary) does Job B's annual salary first exceed Job A's?

Use: $30000 \cdot (1.08)^t > 40000 + 2000t$ . Check $t = 14$ : Job A = $68,000$ . Job B $= 30000 \cdot 1.08^{14} \approx 30000 \times 2.937 \approx 88{,}110$ .
Check $t = 10$ : Job A = $60,000$ . Job B $\approx 30000 \times 2.159 \approx 64{,}770$ .
Check $t = 8$ : Job A = $56,000$ . Job B $\approx 30000 \times 1.851 \approx 55{,}530$ .
Check $t = 9$ : Job A = $58,000$ . Job B $\approx 30000 \times 1.999 \approx 59{,}970$ .

2.

Plan A: $10 per day. Plan B: starts at $0.01 on day 1 and doubles every day.

1.

How much does Plan B pay on day 10?

2.

How much does Plan B pay on day 20? (Use $2^{19} = 524{,}288$ .)

3.

A student argues: "Plan A pays $10 per day, so after 30 days it has paid out $300 total. Plan B starts tiny and I don't see how it can possibly pay more than $300."

Explain why the student's reasoning is flawed, using the concept of exponential domination over linear growth. Support your answer with the payment on day 30.

E

Error Analysis

1.

A student wrote: "I compared $f(x) = 2^x$ and $g(x) = 1000x$ at $x = 1$ through $x = 10$ and found $g$ is always larger. Therefore, $g(x) > f(x)$ for all positive $x$ ."

What is the error in the student's conclusion?

A.

The student correctly found $g > f$ for small $x$ , but incorrectly generalized this to ALL $x$ . The exponential eventually overtakes any linear function.

B.

The student made an arithmetic error — $2^x$ is larger than $1000x$ for all positive $x$ .

C.

The comparison only makes sense for integer $x$ values.

D.

The student is correct — $g(x) = 1000x$ is always larger because its coefficient (1000) is bigger than the base (2).

2.

A student wrote: "I compared $x^2$ and $2^x$ and noticed they are equal at $x = 2$ and $x = 4$ . Therefore $x^2$ is exponential, since it can keep up with $2^x$ ."

What is the error?

A.

Sharing some equal values or having a crossover does not make $x^2$ exponential. After $x = 4$ , $2^x$ permanently exceeds $x^2$ and the gap grows without bound — quadratic cannot keep up.

B.

The student is right — $x^2$ and $2^x$ are equal at two points, so they must be the same type of function.

C.

The error is that $x^2$ and $2^x$ are never equal — the student computed wrong.

D.

Equal values at two points would mean both functions are linear.

F

Challenge

1.

A student says: "I heard that exponential growth always beats polynomial growth, but I found that $1.001^x < x^5$ at $x = 1000$ . Doesn't that disprove the rule?"

Respond to the student. Include an explanation of why the principle is true despite this observation.

2.