The graph of an exponential process

Science and society, grade 11 (3 units) Lesson 5 of 16Learning

The graph of an exponential process

The graph of exponential growth

In the graph of an exponential process the time is $\displaystyle x$ and the amount is $\displaystyle y$ , so its formula is $\displaystyle {y}={M_0\cdot q^{x}}$ .

The graph: the number of bacteria in a dish, doubling every hour – $\displaystyle {y}={100\cdot2^{x}}$ .

\displaystyle y=100\cdot2^{x}

\displaystyle (0,100)

\displaystyle (1,200)

\displaystyle (2,400)

\displaystyle (3,800)

The graph meets the $\displaystyle y$ -axis at the point $\displaystyle (0,\ 100)$ : when $\displaystyle {x}={0}$ , $\displaystyle {y}={M_0}$ – the amount at the start.
$\displaystyle {q}={2}$ is greater than 1, so the graph rises.
The graph rises faster and faster: from hour 0 to hour 1 it rises by 100, and from hour 2 to hour 3 by 400.

The graph of exponential decay

When $\displaystyle q$ is between 0 and 1, the graph falls.

The graph: the percentage of a medicine left in the blood, shrinking by 20% every hour – $\displaystyle {y}={100\cdot0.8^{x}}$ .

\displaystyle y=100\cdot0.8^{x}

\displaystyle (0,100)

The graph falls fast at first, and more and more slowly after.
It approaches the $\displaystyle x$ -axis but never touches it: the medicine never reaches 0.

A time read from the graph

To find when the amount reaches a value, draw a horizontal straight line at the height of the value, and look for where it meets the graph.

When will there be 600 bacteria?

\displaystyle y=100\cdot2^{x}

\displaystyle y=600

\displaystyle (2.585,600)

The line $\displaystyle {y}={600}$ meets the graph above an $\displaystyle x$ between 2 and 3, nearer to 3: about 2.6.
So about two and a half hours from the start there will be 600 bacteria.

A time found by trying values

Without a graph, the time is found by putting values of $\displaystyle t$ into the formula until the amount passes the value.

An area has 500 deer.
The number of deer grows by 20% every year.
When will there be more than 2,000 deer?

Years $\displaystyle t$	6	7	8
$\displaystyle 500\cdot1.2^{t}$	1,492.99	1,791.59	2,149.91

After 7 years there are not yet 2,000 deer, and after 8 years there are.
The number of deer passes 2,000 during the eighth year.

Exercise 1

The graph: the number of cells in a culture, each hour from the start of the experiment – $\displaystyle {y}={50\cdot2^{x}}$ .

Mark the point where the graph meets the $\displaystyle y$ -axis, and the point where the culture has 400 cells.

\displaystyle y=50\cdot2^{x}

\displaystyle y=400

Exercise 2

The graph: the amount of a radioactive substance, in grams, each day – $\displaystyle {y}={400\cdot0.5^{x}}$ .

Mark the point where 50 grams are left.

\displaystyle y=400\cdot0.5^{x}

\displaystyle y=50

Exercise 3

The graph shows an exponential process $\displaystyle {y}={M_0\cdot q^{x}}$ , with the points $\displaystyle A$ and $\displaystyle B$ on it.

Read $\displaystyle M_0$ from the graph, work out $\displaystyle q$ , and write the formula of the graph.

\displaystyle y=\,?

\displaystyle A=(0,200)

\displaystyle B=(1,300)

Exercise 4

A small town has 6,000 residents.
The number of residents grows by 5% every year.

How many residents will there be after 5 years? Round to a whole number.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many residents will there be after 6 years? Round to a whole number.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

In which year will the number of residents pass 8,000?

Exercise 5

A patient’s blood holds 400 mg of a medicine.
Every hour the amount in the blood shrinks by 20%.
Below 100 mg the medicine no longer works.

How many mg will be in the blood after 6 hours? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many mg will be in the blood after 7 hours? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

After how many whole hours does the medicine no longer work?

Formulas: science and society, grade 11 (3 units)

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Growth and decay

An exponential process – $\displaystyle M_0$ the amount at the start, $\displaystyle q$ the factor the amount is multiplied by in each unit of time, $\displaystyle M_t$ the amount after $\displaystyle t$ units of time
$\displaystyle M_t=M_0\cdot q^{t}$
The growth factor – a growth of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1+\frac{p}{100}$
The decay factor – a decay of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1-\frac{p}{100}$
Decay by half-life – $\displaystyle n$ the number of half-lives gone by
$\displaystyle M_t=M_0\cdot\left(\frac{1}{2}\right)^{n}$
The number of half-lives – $\displaystyle t$ the time gone by, $\displaystyle T$ the half-life
$\displaystyle n=\frac{t}{T}$
Linear growth – the amount grows by $\displaystyle d$ in each unit of time
$\displaystyle M_t=M_0+d\cdot t$

Measures of spread

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The range – the greatest datum minus the least
$\displaystyle x_{\max}-x_{\min}$
The deviation of the datum $\displaystyle x_i$ from the mean
$\displaystyle x_i-\bar{x}$
The standard deviation of $\displaystyle n$ data
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}}$
The standard deviation from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \sigma=\sqrt{\frac{f_1(x_1-\bar{x})^2+f_2(x_2-\bar{x})^2+\dots+f_k(x_k-\bar{x})^2}{f_1+f_2+\dots+f_k}}$

Probability

The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities on its branches
$\displaystyle P=p_1\cdot p_2\cdot p_3$
“At least once” in $\displaystyle n$ independent experiments – $\displaystyle p$ the probability that it happens in each
$\displaystyle 1-(1-p)^{n}$
Conditional probability – the probability of $\displaystyle A$ when $\displaystyle B$ is known to have happened; $\displaystyle A\cap B$ – both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
The probability of “both $\displaystyle B$ and $\displaystyle A$ ”
$\displaystyle P(A\cap B)=P(B)\cdot P(A|B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)=P(A)$
The expected value – $\displaystyle x_1, \dots, x_k$ the possible values, $\displaystyle p_1, \dots, p_k$ their probabilities
$\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k$