Task Sets: Construct linear and exponential functions - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

1.

A linear function passes through $(1, 5)$ and $(4, 14)$ . What is the slope?

2.

A table shows $f(0) = 4$ , $f(1) = 12$ , $f(2) = 36$ . What is the common ratio?

3.

The arithmetic sequence $3, 7, 11, 15, \ldots$ corresponds to which linear function?

A.

$f(n) = 3n + 4$ for $n = 1, 2, 3, \ldots$

B.

$f(n) = 4n - 1$ for $n = 1, 2, 3, \ldots$

C.

$f(n) = 3 \cdot 4^{n-1}$ for $n = 1, 2, 3, \ldots$

D.

$f(n) = 4n + 3$ for $n = 1, 2, 3, \ldots$

B

Fluency Practice

1.

A linear function passes through $(2, 7)$ and $(5, 16)$ .

Find the y-intercept $b$ (the value of the function when $x = 0$ ).

2.

The arithmetic sequence $5, 8, 11, 14, \ldots$ corresponds to a linear function $f(n) = mn + b$ for positive integers $n$ .

What is the value of $b$ (the y-intercept)?

3.

An exponential function passes through $(1, 6)$ and $(3, 54)$ .

Find the base $b$ of the function $f(x) = a \cdot b^x$ .

4.

A table of values is shown:

$x$	$g(x)$
2	12
3	36
4	108

Find the initial value $a$ for the exponential function $g(x) = a \cdot b^x$ .

5.

A graph of a linear function passes through integer coordinate points $(1, 5)$ and $(3, 9)$ . Which function matches the graph?

A.

$f(x) = 2x + 3$

B.

$f(x) = 3x + 2$

C.

$f(x) = 2x + 5$

D.

$f(x) = 4x + 1$

C

Mixed Practice

1.

The table below shows input-output pairs of a function. Is the function linear or exponential?

$x$	$h(x)$
0	3
1	6
2	12
3	24

A.

Linear — the outputs increase by 3 each time.

B.

Exponential — the ratios of consecutive outputs are constant (equal to 2).

C.

Linear — the inputs increase by 1 each time.

D.

Exponential — the outputs are getting larger.

2.

A table starts at $x = 3$ , not $x = 0$ :

$x$	$f(x)$
3	24
4	31
5	38
6	45

A student says the y-intercept is $f(3) = 24$ . Is the student correct?

A.

Yes — the first value in the table is always the y-intercept.

B.

No — the y-intercept is $f(0)$ , not $f(3)$ . Since $x = 0$ is not in the table, we must calculate $f(0)$ using the slope.

C.

Yes — since the function is linear, the y-intercept is the initial table value.

D.

It cannot be determined from the table.

3.

An exponential function passes through $(2, 18)$ and $(4, 162)$ .

Find the initial value $a$ (i.e., the value when $x = 0$ ).

4.

A verbal description says: "A pool starts with 200 gallons and gains 50 gallons per minute."

This is a ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ (linear/exponential) function.
The slope is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .
The y-intercept is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .
The function is $W(t) = \_\_\_ t + \_\_\_$ .

function type:

slope:

y-intercept:

slope in W(t):

y-intercept in W(t):

5.

A verbal description says: "A bacteria culture starts with 100 cells and triples every hour."

What is the number of cells after 4 hours?

D

Word Problems

1.

A candle is 20 cm tall when lit and 14 cm tall after 3 hours of burning.

Assuming constant rate, what is the height of the candle after 7 hours?

2.

A bacteria colony has 200 cells at hour 0 and 800 cells at hour 2.

1.

Find the base $b$ of the exponential function $N(t) = a \cdot b^t$ .

2.

How many cells will there be after 5 hours?

3.

An investment is worth $1,000 at the end of year 2 and $4,000 at the end of year 4.

(a) Determine whether the investment grows linearly or exponentially, using the given data.
(b) Write the function and predict the value at the end of year 6.

E

Error Analysis

1.

A student was given the table and concluded the function is exponential:

$x$	$f(x)$
0	2
1	5
2	8
3	11

Student work: "The differences are 3, 3, 3. I'll use these as my 'ratios.' The function is $f(x) = 2 \cdot 3^x$ ."

What is the error in the student's work?

A.

The student found constant differences (confirming linear) but incorrectly concluded the function is exponential and used subtraction results as a base.

B.

The student computed the differences incorrectly — they should be 3, 3, 3.

C.

The student should have used $f(x) = 3x + 2$ instead, which is linear.

D.

The function $f(x) = 2 \cdot 3^x$ is actually correct — check: $f(0) = 2$ , $f(1) = 6$ .

2.

A student solved the following:

"An exponential function grows by 20% per year. Initial value is 500. Write the function."

Student work: $f(t) = 500 \cdot (0.20)^t$

What is the error?

A.

The student forgot the initial value.

B.

The student used the percent rate (0.20) as the base instead of the growth factor (1.20).

C.

The student should have used a negative exponent for growth.

D.

The function is correct — 0.20 is the growth rate.

F

Challenge

1.

The geometric sequence $4, 12, 36, 108, \ldots$ can be described as both a sequence and as a function.

(a) Write the explicit formula for the sequence using sequence notation $a_n = a_1 \cdot r^{n-1}$ .
(b) Rewrite the same function in the form $f(n) = a \cdot b^n$ (function notation).
(c) What is the relationship between $a_1$ and $a$ ?

2.

An exponential function passes through $(3, 24)$ and $(5, 96)$ .