Linear and exponential functions

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1. Linear or exponential: equal differences or equal factors

Warm-up

Three quick questions from earlier lessons.

Discover

Offer A: 1,000 dollars a day for 30 days. Offer B: 1 cent the first day, then double every day. Which pays more on day 30?

  1. Offer B
  2. Offer A

Key idea

Linear adds, exponential multiplies.

Over equal steps in xx, a linear function changes by equal differences (add mm); an exponential function changes by equal factors (multiply by bb). A growing exponential eventually overtakes any linear function.

Worked example

yy: 3, 6, 12, 24.

Not linear
Differences 3, 6, 12: not equal.
Exponential, factor 2
Ratios 2, 2, 2.
24⋅2=4824 \cdot 2 = 48
Next value.

Your turn, guided

Your turn: an exponential function has g(0)=5g(0) = 5, g(1)=15g(1) = 15. Find g(2)g(2).

4545
Factor 3.

Practice ★

  1. A table gives x=0,1,2,3x = 0, 1, 2, 3 and y=4,12,36,108y = 4, 12, 36, 108. What kind of relationship is it?
    1. Exponential: yy is multiplied by 3 at each step
    2. Linear: yy goes up by 8 at each step
    3. Neither
  2. A table gives x=0,1,2,3x = 0, 1, 2, 3 and y=8,24,72,216y = 8, 24, 72, 216. What kind of relationship is it?
    1. Exponential: yy is multiplied by 3 at each step
    2. Linear: yy goes up by 16 at each step
    3. Neither
  1. An exponential function has g(0)=9g(0) = 9 and g(1)=18g(1) = 18. Find g(3)g(3).

Watch out!

Practice ★★

  1. f(x)=57x+380f(x) = 57x + 380 and g(x)=1.04xg(x) = 1.04^x. For very large xx, which is greater?
    1. gg: exponential growth eventually overtakes linear growth
    2. ff: it is greater now and grows faster
    3. They stay about equal
  2. f(x)=73x+397f(x) = 73x + 397 and g(x)=1.04xg(x) = 1.04^x. For very large xx, which is greater?
    1. gg: exponential growth eventually overtakes linear growth
    2. ff: it is greater now and grows faster
    3. They stay about equal

Exit ticket

Three questions to check you have got it.

  1. An exponential function has g(0)=9g(0) = 9 and g(1)=18g(1) = 18. Find g(3)g(3).
  2. An exponential function has g(0)=4g(0) = 4 and g(1)=20g(1) = 20. Find g(3)g(3).
  3. An exponential function has g(0)=5g(0) = 5 and g(1)=15g(1) = 15. Find g(3)g(3).

Summary card

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

2. Exponential growth and decay: multiplying by a constant factor

Discover

Two ponds each start with 5 m² of algae. In pond A, the area grows by 5 m² each week. In pond B, the area doubles each week.

After 1 week both ponds have 10 m². Which pond has more algae after 4 weeks? Guess first.

  1. Pond A
  2. Pond B
  3. They have the same

Then check with a small table for weeks 0 to 4.

Key idea

When a quantity is multiplied by the same factor bb at each step, it changes exponentially.

Start aa, factor bb each step: after nn steps the amount is a⋅bna \cdot b^n

  • If b>1b > 1, the quantity grows (exponential growth).
  • If 0<b<10 < b < 1, it shrinks (exponential decay).
  • Losing p%p\% each step keeps (100−p)%(100 - p)\%, so the factor is 1−p1001 - \frac{p}{100}. Gaining p%p\% gives the factor 1+p1001 + \frac{p}{100}.

Worked example

A photo is 1000 pixels wide. Each time it is resized to 80% of its width. How wide is it after 3 resizes?

W=1000⋅0.8⋅0.8⋅0.8W = 1000 \cdot 0.8 \cdot 0.8 \cdot 0.8
Keeping 80% means multiplying by 0.8 each time.
W=1000⋅0.83W = 1000 \cdot 0.8^3
Three equal factors make a power.
W=512W = 512 pixels
0.8³ = 0.512.

Your turn, guided

Your turn: a plant is 3 cm tall, and its height triples every week. How tall is it after 4 weeks? Fill in the missing line.

h=3⋅34h = 3 \cdot 3^4
Start at 3 and multiply by 3 four times.
h=243h = 243 cm
3⁴ = 81, and 3 · 81 = 243.

Practice ★

  1. The amount of algae starts at 5 m² and is multiplied by 4 each week. Complete the table.
  2. The amount of algae starts at 2 m² and is multiplied by 2 each week. Complete the table.
  1. A culture starts with 7 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 3 weeks?
  2. A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 5 weeks?
  3. A culture starts with 3 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 5 weeks?
  4. A culture starts with 6 grams of algae. Each week, the amount is multiplied by 5. How many grams are there after 5 weeks?
  5. A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 3 weeks?

Watch out!

Practice ★★

  1. A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width W(t)W(t) after tt resizes.
  2. A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width W(t)W(t) after tt resizes.
  3. A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width W(t)W(t) after tt resizes.
  4. A photo is 8000 pixels wide. Each time it is resized, its width loses 10%. Write the width W(t)W(t) after tt resizes.
  5. A photo is 4000 pixels wide. Each time it is resized, its width loses 30%. Write the width W(t)W(t) after tt resizes.

Exit ticket

Three questions to check you have got it.

  1. A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width W(t)W(t) after tt resizes.
  2. A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width W(t)W(t) after tt resizes.
  3. A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width W(t)W(t) after tt resizes.

Summary card

Exponential change: the same factor at every step

  • Start aa, factor bb per step: after nn steps, a⋅bna \cdot b^n.
  • Gain p%p\%: factor 1+p1001 + \frac{p}{100}. Lose p%p\%: factor 1−p1001 - \frac{p}{100}.

Watch out: adding the same amount each step is linear, not exponential.

3. Interpret exponential functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

f(x)=40⋅(12)xf(x) = 40 \cdot \left(\tfrac{1}{2}\right)^x. What is f(0)f(0)?

  1. 4040
  2. 00

Key idea

Read aa and bb from f(x)=a⋅bxf(x) = a \cdot b^x.

a=f(0)a = f(0) is the starting value (the yy-intercept). bb is the growth factor: b>1b > 1 growth, 0<b<10 < b < 1 decay. Negative inputs go back in time: f(−1)=a÷bf(-1) = a \div b.

Worked example

Graph through (0,3)(0, 3) and (1,12)(1, 12).

a=3a = 3
Starting value.
b=4b = 4
12 ÷ 3.
f(−1)=3÷4=0.75f(-1) = 3 \div 4 = 0.75
One step back.

Your turn, guided

Your turn: f(x)=50⋅5xf(x) = 50 \cdot 5^x. Find f(−1)f(-1).

1010
50 ÷ 5.

Practice ★

  1. f(x)=24⋅4xf(x) = 24 \cdot 4^x. Find f(−1)f(-1).
  2. f(x)=18⋅3xf(x) = 18 \cdot 3^x. Find f(−1)f(-1).
  1. Is f(x)=4⋅(14)xf(x) = 4 \cdot \left(\tfrac{1}{ 4 }\right)^x exponential growth or decay?
    1. Decay: each step multiplies by 14\tfrac{1}{ 4 }
    2. Growth, because 4 is greater than 1
    3. A linear decrease

Watch out!

Practice ★★

  1. The graph of an exponential function passes through (0,8)(0, 8) and (1,16)(1, 16). Write f(x)f(x).
  2. The graph of an exponential function passes through (0,7)(0, 7) and (1,21)(1, 21). Write f(x)f(x).

Exit ticket

Three questions to check you have got it.

  1. f(x)=24⋅4xf(x) = 24 \cdot 4^x. Find f(−1)f(-1).
  2. f(x)=18⋅3xf(x) = 18 \cdot 3^x. Find f(−1)f(-1).
  3. f(x)=27⋅3xf(x) = 27 \cdot 3^x. Find f(−1)f(-1).

Summary card

Reading exponential functions

aa: value at x=0x = 0; bb: factor per step; b−1=1/bb^{-1} = 1/b.

Watch out: negative exponent, positive value.

4. Write an exponential function from a description or two points

Warm-up

Three quick questions from earlier lessons.

Discover

A table shows a function: f(0)=4f(0) = 4, f(1)=12f(1) = 12, f(2)=36f(2) = 36, f(3)=108f(3) = 108.

What do you multiply by to go from one value to the next? Can you write one formula that gives all four values?

Key idea

An exponential function can be written

f(x)=a⋅bxf(x) = a \cdot b^x, where a=f(0)a = f(0) and bb is the factor for each increase of 1 in xx

If the quantity is multiplied by bb every nn units instead, divide the input by nn: f(x)=a⋅bxnf(x) = a \cdot b^{\frac{x}{n}}.

Worked example

Write ff for the table above.

a=4a = 4
The starting value is f(0).
b=3b = 3
Divide consecutive values: 12 ÷ 4 = 3.
f(x)=4⋅3xf(x) = 4 \cdot 3^x
Put them in the form a · b^x.

Your turn, guided

Your turn: f(0)=5f(0) = 5 and f(1)=10f(1) = 10. Fill in the missing line.

a=5a = 5, b=2b = 2
10 ÷ 5 = 2.
f(x)=5⋅2xf(x) = 5 \cdot 2^x
Use the form a · b^x.

Practice ★

  1. Plot the points of the graph of f(x)=4⋅2xf(x) = 4 \cdot 2^x at x=0x = 0 and x=1x = 1.
  2. Plot the points of the graph of f(x)=3⋅4xf(x) = 3 \cdot 4^x at x=0x = 0 and x=1x = 1.
  1. An exponential function ff has f(0)=9f(0) = 9, and each time xx increases by 1, the value of ff is multiplied by 5. Write f(x)f(x).
  2. An exponential function ff has f(0)=9f(0) = 9, and each time xx increases by 1, the value of ff is multiplied by 4. Write f(x)f(x).
  3. An exponential function ff has f(0)=6f(0) = 6, and each time xx increases by 1, the value of ff is multiplied by 2. Write f(x)f(x).
  4. An exponential function ff has f(0)=6f(0) = 6, and each time xx increases by 1, the value of ff is multiplied by 4. Write f(x)f(x).
  5. An exponential function ff has f(0)=4f(0) = 4, and each time xx increases by 1, the value of ff is multiplied by 2. Write f(x)f(x).

Watch out!

Practice ★★

  1. A population is 7 thousand at t=0t = 0 and is multiplied by 3 every 4 years. Write the population P(t)P(t), in thousands, after tt years.
  2. A population is 8 thousand at t=0t = 0 and is multiplied by 3 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  3. A population is 4 thousand at t=0t = 0 and is multiplied by 2 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  4. A population is 8 thousand at t=0t = 0 and is multiplied by 2 every 2 years. Write the population P(t)P(t), in thousands, after tt years.
  5. A population is 9 thousand at t=0t = 0 and is multiplied by 3 every 5 years. Write the population P(t)P(t), in thousands, after tt years.

Exit ticket

Three questions to check you have got it.

  1. A population is 7 thousand at t=0t = 0 and is multiplied by 3 every 4 years. Write the population P(t)P(t), in thousands, after tt years.
  2. A population is 8 thousand at t=0t = 0 and is multiplied by 3 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  3. A population is 4 thousand at t=0t = 0 and is multiplied by 2 every 3 years. Write the population P(t)P(t), in thousands, after tt years.

Summary card

Writing an exponential function

  • f(x)=a⋅bxf(x) = a \cdot b^x: a=f(0)a = f(0), b=f(1)f(0)b = \dfrac{f(1)}{f(0)}.
  • Multiplied by bb every nn units: f(x)=a⋅bxnf(x) = a \cdot b^{\frac{x}{n}}.

Watch out: the starting value is f(0)f(0), and it goes in front, not in the base.

5. Percent change and compound interest

Warm-up

Three quick questions from earlier lessons.

Discover

A 50-dollar game goes up by 10%. Which calculation gives the new price in one step?

  1. 50⋅1.150 \cdot 1.1
  2. 50⋅0.150 \cdot 0.1

Key idea

Percent change is a growth factor.

An increase of rr (as a decimal) multiplies by 1+r1 + r; a decrease multiplies by 1−r1 - r. Compound interest at yearly rate rr, compounded nn times a year: A=P(1+rn)ntA = P\left(1 + \tfrac{r}{n}\right)^{nt}; yearly: A=P(1+r)tA = P(1 + r)^t.

Worked example

800 dollars at 5% per year, compounded yearly, for 3 years.

1.051.05
Factor.
800⋅1.053800 \cdot 1.05^3
Three years.
≈926.10\approx 926.10 dollars
Calculator.

Your turn, guided

Your turn: the growth factor for a 7% increase.

1.071.07
1 + 0.07.

Practice ★

  1. A quantity grows by 9% each year. By what number is it multiplied each year (the growth factor)?
  2. A quantity grows by 16% each year. By what number is it multiplied each year (the growth factor)?
  1. 882 dollars are invested at 5% per year, compounded yearly. Write the value after 4 years (an exact expression is fine).

Watch out!

Practice ★★

  1. 660 dollars are invested at 4% per year, compounded monthly. Write the value after 2 years (an exact expression is fine).
  2. 1818 dollars are invested at 7% per year, compounded monthly. Write the value after 3 years (an exact expression is fine).
  1. Boss problem. Ana puts 1800 dollars in a savings account that pays 8% per year, compounded yearly.

Exit ticket

Three questions to check you have got it.

  1. A quantity grows by 9% each year. By what number is it multiplied each year (the growth factor)?
  2. A quantity grows by 16% each year. By what number is it multiplied each year (the growth factor)?
  3. A quantity grows by 34% each year. By what number is it multiplied each year (the growth factor)?

Summary card

Percent change and interest

Increase rr: × (1+r)(1 + r); decrease: × (1−r)(1 - r); compounding: rate and exponent per period.

Watch out: 1.1, not 0.1, for a 10% increase.

Answers: Linear and exponential functions

1. Linear or exponential: equal differences or equal factors

A

  1. Exponential: yy is multiplied by 3 at each step
  2. Exponential: yy is multiplied by 3 at each step

B

  1. 7272

C

  1. gg: exponential growth eventually overtakes linear growth
  2. gg: exponential growth eventually overtakes linear growth

D

  1. 7272
  2. 500500
  3. 135135

2. Exponential growth and decay: multiplying by a constant factor

A

  1. 20,80,32020, 80, 320
  2. 4,8,164, 8, 16

B

  1. 189189
  2. 160160
  3. 729729
  4. 1875018750
  5. 4040

C

  1. 8000(35)t8000 \left(\frac{3}{5}\right)^{t}
  2. 2000(710)t2000 \left(\frac{7}{10}\right)^{t}
  3. 2000(34)t2000 \left(\frac{3}{4}\right)^{t}
  4. 8000(910)t8000 \left(\frac{9}{10}\right)^{t}
  5. 4000(710)t4000 \left(\frac{7}{10}\right)^{t}

D

  1. 8000(35)t8000 \left(\frac{3}{5}\right)^{t}
  2. 2000(710)t2000 \left(\frac{7}{10}\right)^{t}
  3. 2000(34)t2000 \left(\frac{3}{4}\right)^{t}

3. Interpret exponential functions and their graphs

A

  1. 66
  2. 66

B

  1. Decay: each step multiplies by 14\tfrac{1}{ 4 }

C

  1. 8⋅2x8 \cdot 2^{x}
  2. 7⋅3x7 \cdot 3^{x}

D

  1. 66
  2. 66
  3. 99

4. Write an exponential function from a description or two points

A

  1. (0,4), (1,8)(0, 4),\ (1, 8)
  2. (0,3), (1,12)(0, 3),\ (1, 12)

B

  1. 9⋅5x9 \cdot 5^{x}
  2. 9⋅4x9 \cdot 4^{x}
  3. 6⋅2x6 \cdot 2^{x}
  4. 6⋅4x6 \cdot 4^{x}
  5. 4⋅2x4 \cdot 2^{x}

C

  1. 7⋅3t47 \cdot 3^{\frac{t}{4}}
  2. 8⋅3t38 \cdot 3^{\frac{t}{3}}
  3. 4⋅2t34 \cdot 2^{\frac{t}{3}}
  4. 8⋅2t28 \cdot 2^{\frac{t}{2}}
  5. 9⋅3t59 \cdot 3^{\frac{t}{5}}

D

  1. 7⋅3t47 \cdot 3^{\frac{t}{4}}
  2. 8⋅3t38 \cdot 3^{\frac{t}{3}}
  3. 4⋅2t34 \cdot 2^{\frac{t}{3}}

5. Percent change and compound interest

A

  1. 109100\frac{109}{100}
  2. 2925\frac{29}{25}

B

  1. 8576612180000\frac{85766121}{80000}

C

  1. 33650272028774857920304120635356211338224302652032980013192114707158941350000000000000000000000000000000000000000000000\frac{3365027202877485792030412063535621133822430265203298001319211}{4707158941350000000000000000000000000000000000000000000000}
  2. 8826389678019491028561666037929750124492491838427862676103233692368874238245665042611416984282293575685580883930139377881943616213632296905944976064512000000000000000000000000000000000000000000000000000000000000000000000000\frac{88263896780194910285616660379297501244924918384278626761032336923688742382456650426114169842822935756855808839301}{39377881943616213632296905944976064512000000000000000000000000000000000000000000000000000000000000000000000000}

D

  1. 5248825\frac{52488}{25}

E

  1. 109100\frac{109}{100}
  2. 2925\frac{29}{25}
  3. 6750\frac{67}{50}