Choosing a model: linear, quadratic or exponential

Science and society, grade 12 (3 units) Lesson 7 of 14Learning

Choosing a model: linear, quadratic or exponential

Three tables, three models

In these three tables $\displaystyle x$ grows by 1 each time. What decides the model is how $\displaystyle y$ changes from column to column:

A taxi ride – the price in shekels after $\displaystyle x$ km:

$\displaystyle x$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$	$\displaystyle 4$
$\displaystyle y$	$\displaystyle 12$	$\displaystyle 15$	$\displaystyle 18$	$\displaystyle 21$	$\displaystyle 24$
Difference		$\displaystyle +3$	$\displaystyle +3$	$\displaystyle +3$	$\displaystyle +3$

A falling stone – how many metres it has fallen after $\displaystyle x$ seconds:

$\displaystyle x$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$	$\displaystyle 4$
$\displaystyle y$	$\displaystyle 0$	$\displaystyle 5$	$\displaystyle 20$	$\displaystyle 45$	$\displaystyle 80$
Difference		$\displaystyle +5$	$\displaystyle +15$	$\displaystyle +25$	$\displaystyle +35$

Bacteria in a dish – the number of bacteria after $\displaystyle x$ hours:

$\displaystyle x$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$	$\displaystyle 4$
$\displaystyle y$	$\displaystyle 100$	$\displaystyle 200$	$\displaystyle 400$	$\displaystyle 800$	$\displaystyle 1600$
Ratio		$\displaystyle \times2$	$\displaystyle \times2$	$\displaystyle \times2$	$\displaystyle \times2$

How to tell the model

When $\displaystyle x$ grows in equal steps:

A linear relation – the differences between neighbouring values of $\displaystyle y$ are equal: in the taxi each km adds 3 ₪. The formula: $\displaystyle {y}={3x+12}$ .
A quadratic relation – the differences are not equal, but each grows by the same number: for the stone $\displaystyle 5, 15, 25, 35$ – each difference is 10 more than the one before. The formula: $\displaystyle {y}={5x^{2}}$ .
An exponential relation – growth or decay, as you learnt in grade 11: the ratio of neighbouring values is fixed: the bacteria double every hour. The formula: $\displaystyle {y}={100\cdot2^{x}}$ .

\displaystyle y=mx+b

\displaystyle y=ax^{2}+bx+c

\displaystyle y=a\cdot b^{x}

Measured braking distances

In a test of a new car the braking distance was measured at different speeds:

Speed (km/h)	$\displaystyle 20$	$\displaystyle 40$	$\displaystyle 60$	$\displaystyle 80$
Distance (metres)	$\displaystyle 2$	$\displaystyle 8$	$\displaystyle 18$	$\displaystyle 32$
Difference		$\displaystyle +6$	$\displaystyle +10$	$\displaystyle +14$

The differences are not equal – it is not a linear relation.
The ratios, $\displaystyle {\frac{8}{2}}={4}$ , $\displaystyle {\frac{18}{8}}={2.25}$ , are not equal – it is not an exponential relation.
Each difference is 4 more than the one before – a quadratic relation.

So at 100 km/h the next difference will be $\displaystyle {14+4}={18}$ , and the braking distance $\displaystyle {32+18}={50}$ metres.

In real measurements the numbers fit the model only roughly: look for the model whose differences or ratios are nearly fixed.

Exercise 1

Which model fits each table: linear, quadratic or exponential? Explain each answer.

$\displaystyle x$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$	$\displaystyle 4$
A	$\displaystyle 3$	$\displaystyle 7$	$\displaystyle 11$	$\displaystyle 15$	$\displaystyle 19$
B	$\displaystyle 2$	$\displaystyle 6$	$\displaystyle 18$	$\displaystyle 54$	$\displaystyle 162$
C	$\displaystyle 1$	$\displaystyle 3$	$\displaystyle 9$	$\displaystyle 19$	$\displaystyle 33$

Exercise 2

The board shows the points of table C of Exercise 1, up to $\displaystyle {x}={3}$ .
Three suggested formulas: $\displaystyle {y}={4x+1}$ , $\displaystyle {y}={3^{x}}$ , $\displaystyle {y}={2x^{2}+1}$
Write each formula on the board, and see its graph. Which formula passes through all the points? Check.

\displaystyle (0,1)

\displaystyle (1,3)

\displaystyle (2,9)

\displaystyle (3,19)

Exercise 3

In a physics lesson experiment a ball was dropped from high up, and the distance it had fallen was measured:

Time (seconds)	$\displaystyle 0$	$\displaystyle 0.5$	$\displaystyle 1$	$\displaystyle 1.5$	$\displaystyle 2$
Distance (metres)	$\displaystyle 0$	$\displaystyle 1.25$	$\displaystyle 5$	$\displaystyle 11.25$	$\displaystyle 20$

Work out the differences. Which model fits the table?

How many metres had the ball fallen after 2.5 seconds, by the model?

/ – 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

Exercise 4

The number of people living in a new town, at the start of each year:

Year	$\displaystyle 2023$	$\displaystyle 2024$	$\displaystyle 2025$	$\displaystyle 2026$
People	$\displaystyle 40000$	$\displaystyle 42000$	$\displaystyle 44100$	$\displaystyle 46305$

The differences are $\displaystyle 2000$ , $\displaystyle 2100$ , $\displaystyle 2205$ . Is it a quadratic relation? Which model fits, and why?

By what number is the town’s population multiplied each year?

/ – 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

Formulas: science and society, grade 12 (3 units)

Open as a page

A quadratic model

A quadratic model – $\displaystyle a<0$ : it has a highest point, $\displaystyle a>0$ : a lowest point
$\displaystyle y=ax^{2}+bx+c$ $\displaystyle a\neq0$
The vertex’s $\displaystyle x$ – the axis of symmetry
$\displaystyle x=-\frac{b}{2a}$
The vertex’s $\displaystyle x$ – midway between two values of $\displaystyle x$ with the same $\displaystyle y$
$\displaystyle x=\frac{x_1+x_2}{2}$
The quadratic formula for $\displaystyle {ax^{2}+bx+c}={0}$ – the zeros
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Other models

A linear relation – the differences in the table are equal
$\displaystyle y=mx+b$
An exponential relation – the ratio of two neighbouring values in the table is fixed, $\displaystyle b$
$\displaystyle y=a\cdot b^{x}$

The normal distribution

The standard deviation of $\displaystyle n$ data – $\displaystyle \bar{x}$ their mean
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}+\dots+(x_n-\bar{x})^{2}}{n}}$
In a normal distribution – $\displaystyle \mu$ the mean, $\displaystyle \sigma$ the standard deviation: about 68% of the data within one standard deviation of the mean (34% on each side)
$\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%$
About 95% within two standard deviations of the mean (another 13.5% on each side)
$\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%$
About 2.5% above $\displaystyle \mu+2\sigma$ , and about 2.5% below $\displaystyle \mu-2\sigma$
$\displaystyle P(x>\mu+2\sigma)\approx2.5\%$
How many of $\displaystyle n$ data are $\displaystyle p$ percent of them
$\displaystyle n\cdot p\%$

The standard score

The standard score of the value $\displaystyle x$ – how many standard deviations it is above the mean (or below it, when $\displaystyle z<0$ )
$\displaystyle z=\frac{x-\mu}{\sigma}$
The value whose standard score is $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$