Prep 3 SS · Mathematics

Functions · Graphical representation of functions

Identifying the Right Function Model

Before you start, make sure you know: Exponential and logarithmic functions (High School Year 2 (semesters 3 and 4))

Before you start, make sure you know: Linear function; inverse; root; piecewise; absolute value (High School Year 2 (semesters 3 and 4))

So far you have worked with functions given by their formula. In real life it often goes the other way around: you have a table of data (sales, users, temperature) and you must choose which family of functions describes it, before calculating any parameter. Imagine that the agua fresca stand at a street market (tianguis) records its sales week by week: will they keep growing at the same rate forever? Choosing the wrong model leads to absurd predictions, and on this page you will learn how to avoid that.

Linear: constant increments

Cost of printing invitation cards

Cards	10	20	30	40
Cost (pesos)	170	290	410	530

Consecutive differences: $290-170=120$, $410-290=120$, $530-410=120$. They are constant.

The model is linear, $C(x)=mx+b$, with $m=\dfrac{120}{10}=12$ pesos per card (we divide by the step in $x$, which is 10).

With $C(10)=170$: $170=12(10)+b$, so $b=50$. Then $C(x)=12x+50$: a fixed cost of $\$50$ and $\$12$ per card.

Practice★☆☆

These data show the cost of printing posters depending on the number of units: | Posters | $10$ | $20$ | $30$ | $40$ | |---|---|---|---|---| | Cost ( pesos) | $340$ |…

These data show the cost of printing posters depending on the number of units:

Posters	$10$	$20$	$30$	$40$
Cost ( pesos)	$340$	$580$	$820$	$1\,060$

a) Check that the differences between consecutive costs are constant.
b) Which functional model is suitable? Determine its expression and calculate how much $100$ posters would cost.

Solve

Practice★☆☆

True or false? Reason your answer with a counterexample: "Any dataset that grows fits a linear model well."

Solve

Quadratic: a cloud of points shaped like a parabola

Practice★★☆

The figure shows three scatter plots corresponding to three different phenomena, as shown in the figure. Match each graph with one of these phenomena and with the…

The figure shows three scatter plots corresponding to three different phenomena, as shown in the figure.

Match each graph with one of these phenomena and with the functional model that best suits it, justifying each choice:

1. The average production cost of a company as a function of the number of units manufactured.
1. The followers of an account that grows through recommendations between users.
1. A family's monthly transportation expenses as a function of the number of trips made.

Solve

Practice★★★

An analyst fits the monthly electricity consumption data of a municipality (which is high in winter, low in spring and rises again in summer due to air conditioning)…

An analyst fits the monthly electricity consumption data of a municipality (which is high in winter, low in spring and rises again in summer due to air conditioning) with a second-degree polynomial.

a) Reason why a quadratic model can yield absurd medium-term predictions for this phenomenon.
b) Which family of functions would be more suitable for a phenomenon that repeats every year? Write down the general form of that model and explain the meaning of its parameters.
c) What general precaution must be taken when extrapolating any model fitted to data?

Solve

Exponential: growth that speeds up

Practice★★☆

The population of a small village with limited access to housing has gone from $800$ to $1,600$ inhabitants in ten years, but the rate of growth is slowing down as…

The population of a small village with limited access to housing has gone from $800$ to $1,600$ inhabitants in ten years, but the rate of growth is slowing down as available housing is occupied (estimated maximum capacity: $2,000$ inhabitants).

a) Explain why a pure exponential model would not be realistic in the long term.
b) Which model with a horizontal asymptote would be more suitable? Describe its shape and its qualitative parameters.
c) What $30$-year prediction would each model give? Comment on the difference.

Solve

Practice★★★

The weekly cases of a flu epidemic in a district were: | Week | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | |---|---|---|---|---|---|---|---|---| | Cases | $40$ |…

The weekly cases of a flu epidemic in a district were:

Week	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Cases	$40$	$95$	$210$	$380$	$430$	$350$	$210$	$90$

a) Plot the data and reason why neither an exponential nor a logistic model describes the whole process.
b) Propose what type of model (or combination of piecewise models) could fit each phase of the phenomenon, justifying your choice.
c) What reasonable prediction would you make for week $10$?

You can do it on paper and send a photo with your answer.

Solve

Exponentials and logarithms with different bases

The base $b$ of an exponential $y=b^x$ changes its shape a lot. Compare the bases $2$ and $3$ (they grow) and $\tfrac12$ (it decays). All of them pass through $(0, 1)$ and never touch the $x$-axis.

Point of $y=2^x$	$(-2, \tfrac14)$	$(-1, \tfrac12)$	$(0, 1)$	$(1, 2)$	$(2, 4)$	$(3, 8)$
Point of $y=\log_2 x$	$(\tfrac14, -2)$	$(\tfrac12, -1)$	$(1, 0)$	$(2, 1)$	$(4, 2)$	$(8, 3)$

The curves $y=2^x$, $y=\log_2 x$, and the line $y=x$ look like this: the logarithm is the "mirror" of the exponential.

With another base the mirror still holds. Here, $y=3^x$ and $y=\log_3 x$:

Let's practice! 4 easy exercises in this section

★☆☆ The graph shows the monthly user data for a podcast platform during its first year. Which…

Logarithmic and saturation-ceiling models

Cooling of a coffee (room temperature: 22 °C)

$t$ (min)	0	5	10	15	20	25
$T$ (°C)	90	72	59	50	44	40

The differences ($-18, -13, -9, -6, -4$) are not constant, so it is not linear. The temperature approaches 22 °C more and more slowly without crossing it.

What really cools down is the difference from the room temperature: $T-22 = 68, 50, 37, 28, 22, 18$. It shrinks by an almost constant factor (about $0.75$ every 5 minutes), so it is exponential. The model is $T(t)=22+b\,e^{-kt}$, with a horizontal asymptote at $y=22$, not at $y=0$.

Parameters: at $t=0$, $T-22=68$, so $b=68$. With $t=5$: $68e^{-5k}=50$, so $e^{-5k}=\dfrac{50}{68}$ and $k=\dfrac{\ln(68/50)}{5}\approx 0.06$.

Model: $T(t)=22+68e^{-0.06t}$. Check: $T(5)\approx 72.4$ °C and $T(10)\approx 59.3$ °C, very close to the table. The last data points drift a bit (at $t=25$ it gives $\approx 37$ versus 40): a model describes, it does not reproduce every data point.

Fitting the model with technology

You do not have to get $b$ and $k$ by hand. With the coffee data:

GeoGebra: in the Spreadsheet View, enter $t$ in column A and $T-22$ in column B (that is, $68, 50, 37, 28, 22, 18$). Select both, choose "Create List of Points," and type FitExp(list1) in the input bar. You get a function $a\,e^{bx}$ with $a\approx 68$ and $b\approx -0.06$. Add $22$ to it to get $T(t)$.
Spreadsheet (Excel, Google Sheets): enter the columns $t$ and $T-22$, insert a scatter plot, add an exponential trendline, and turn on "display equation" and $R^2$. The closer $R^2$ is to $1$, the better the fit.
Also try the linear trendline: you will see that $R^2$ is lower and the curve separates from the points. Technology helps you compare models, but the judgment of choosing one (does it have a ceiling? does it repeat?) is still yours.

Practice★☆☆

The graph shows the monthly user data for a podcast platform during its first year. Which of these functional models best describes the trend: linear, quadratic,…

The graph shows the monthly user data for a podcast platform during its first year.

Which of these functional models best describes the trend: linear, quadratic, exponential or logarithmic? Justify your choice by ruling out at least two of the other models.

Solve

Practice★☆☆

The temperature of a cup of coffee is measured every $5$ minutes after serving: | $t$ (min) | $0$ | $5$ | $10$ | $15$ | $20$ | $25$ | |---|---|---|---|---|---|---| | $T$…

The temperature of a cup of coffee is measured every $5$ minutes after serving:

$t$ (min)	$0$	$5$	$10$	$15$	$20$	$25$
$T$ ($^\circ$C)	$90$	$72$	$59$	$50$	$44$	$40$

a) Plot the points and explain why a linear model is not suitable.
b) Given that the room temperature is $22\ ^\circ$C, reason which type of model (with which asymptote) best describes the cooling.

You can do it on paper and send a photo with your answer.

Solve

Choosing between models and the risks of extrapolating

Back to the beginning: the agua fresca stand cannot keep growing at the same rate forever. If its sales go up more and more slowly toward a maximum (the number of people who walk through the tianguis), the model is one with a ceiling, not linear or exponential.

Let's practice! 2 medium exercises in this section

★★☆ The figure shows three scatter plots corresponding to three different phenomena, as shown…★★☆ The population of a small village with limited access to housing has gone from $800$ to…

Summary

Linear: constant simple differences.
Quadratic: parabola shape and constant second differences; only reliable in the observed stretch.
Exponential $y=a\,b^x$: values are multiplied by a fixed factor; it grows if $b>1$ and decays if $0
Logarithmic $y=\log_b x$: inverse of the exponential (graph reflected across $y=x$); it grows without a ceiling but more and more slowly.
With a saturation ceiling (asymptote at $y=M$): population with limited capacity, cooling toward room temperature ($T=22+b\,e^{-kt}$), spread of a rumor.
Periodic: phenomena that repeat (annual electricity use); a polynomial does not describe them.
GeoGebra or a spreadsheet help you fit and compare models, but the choice still depends on the phenomenon.
When extrapolating far from the data, always check whether the result makes sense.

Let's practice! 2 hard exercises in this section

★★★ The weekly cases of a flu epidemic in a district were: | Week | $1$ | $2$ | $3$ | $4$ |…★★★ An analyst fits the monthly electricity consumption data of a municipality (which is high…

See all the exercises in this section