Topic 1.14 · CED: Function Model Construction and Application

Function Model Construction

10 MIN READ9 IDEAS23 PROBLEMS8 flashcards

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01
Once a regression is stored in Y1, you can find any predicted value quickly from the home screen — just type Y1(the x-value) and press enter.
02
Residual = Actual Value − Predicted Value

Practice 23 problems Next: Chapter 2.1 Change in Arithmetic and Geometric Sequences

Building a Regression Model on a Calculator (TI-84)

Building a regression model on a graphing calculator takes two steps: entering the data, then selecting the regression type. Here's the exact sequence on a TI-84 — including which buttons to press and what the screen shows at each stage.

Four keys do the work: STAT to start, 2nd and ALPHA to type list and variable names, and ENTER to confirm. The arrow keys move between menu items.

CONCEPT

Step 1 — Enter the Data

Press STAT, then select 1: Edit... from the menu.

Enter your x-values into list L1 and your y-values into the matching list L2.

What the screen looks like after entering data — x-values in L1, y-values in L2. The highlighted cell shows where you're currently typing.

CONCEPT

Step 2 — Run the Regression

Press STAT again, then arrow RIGHT to the CALC menu.

Select the regression that matches your model type: 4:LinReg(ax+b), 5:QuadReg, 6:CubicReg, 7:QuartReg, or 8:LinReg(a+bx).

Set Xlist: L1 and Ylist: L2 (L1 and L2 are the blue labels above the "1" and "2" keys — press 2nd then 1 or 2 to type them).

For Store RegEQ:, enter Y1 (press ALPHA then TRACE to type Y1). This saves the model so you can reuse it later.

Highlight Calculate and press ENTER to get your equation.

The CALC menu — arrow down to the regression type you need (here, 5:QuadReg is highlighted) and press ENTER.

After pressing Calculate, the screen shows your model's coefficients — matching the data above exactly: $a=1$ , $b=5$ , $c=0$ (this data was built from $y=x^{2}+5x$ , so $R^{2}=1$ , a perfect fit).

CONCEPT

A Note on the Two Linear Options

The TI-84 has two linear regressions: 4:LinReg(ax+b) and 8:LinReg(a+bx). They're mathematically equivalent — just written with $a$ and $b$ swapped. AP Precalculus (matching AP Stats) generally uses option 8, $y=a$ +bx.

KEY RULE

Once a regression is stored in Y1, you can find any predicted value quickly from the home screen — just type Y1(the x-value) and press ENTER.

COMMON MISTAKE

Using a DIFFERENT calculator (Casio, a different TI model, an online tool)? The exact button sequence will differ from what's shown here. Search "[your calculator model] + regression" (e.g. "Casio fx-9750 quadratic regression") to find the right steps for your device — the CONCEPT (enter data, choose model type, run it) stays the same everywhere.

Constructing a Polynomial Model from Data

Once you've selected a model TYPE (1.13), the next step is actually constructing it — finding the specific coefficients that fit your data, usually with a calculator's regression feature.

Worked example

Electric vehicle sales (thousands) in a small country are tracked over several years. Model $N(t)$ = at³+bt²+ct+d, and use it to find the average rate of change from year 2 to year 10.

year	0	2	4	6	8	10	12
sales (thousands)	5	11	7	3	20	55	105

The scatter plot dips before shooting up — a classic cubic shape.

N(t) = at^{3} + bt^{2} + ct + d

01
Enter the data into a calculator's cubic regression feature (not solved by hand — 7 points, 4 unknowns).
02
The regression gives:
$a \approx 0.149, \quad b \approx -1.327, \quad c \approx 2.784, \quad d \approx 6.5$
03
Use the model to find average rate of change from $t=2$ to $t=10$ :
$\frac{N(10) - N(2)}{10 - 2} \approx 5.37 \text{ (thousand/year)}$
04
In context: between year 2 and year 10, EV sales grew at an average rate of about 5.37 thousand vehicles per year.

COMMON MISTAKE

Regression coefficients from real data are almost never clean integers — round sensibly (usually 3 decimal places) and carry the rounded model forward consistently.

Always double check which regression type matches your data's shape BEFORE running it (1.13's tools apply here too).

Types of Regression — a Visual Reference

Your calculator can fit many more shapes than just polynomials. Recognizing these shapes helps you pick the right regression before you even touch a button.

Eight common regression families — linear, quadratic, cubic, quartic, exponential, logarithmic, logistic, and sine.

Inversely Proportional Models

CONCEPT

The Basic Form

Two quantities are inversely proportional if their PRODUCT is always the same constant $k$ .

y = \frac{k}{x} \quad \Longleftrightarrow \quad x \cdot y = k

Worked example

The time to complete a landscaping job is inversely proportional to the number of workers assigned. With 4 workers, the job takes 15 hours. How long would it take with 10 workers?

01
Set up the model and solve for $k$ using the given point:
$4 \times 15 = 60 = k$
02
Use $k$ to find the time with 10 workers:
$\text{time} = \frac{60}{10} = 6 \text{ hours}$

Some quantities are inversely proportional to the SQUARE of another (common in physics — light, sound, and gravity all spread out this way). Here, it's $x^{2}$ times $y$ that stays constant, NOT $x$ times $y$ .

y = \frac{k}{x^{2}} \quad \Longleftrightarrow \quad x^{2} \cdot y = k

Worked example

The loudness of a sound (in decibels) is inversely proportional to the SQUARE of the distance from the source. At 3 meters, the loudness is 80 dB. Find the loudness at 6 meters.

I = \frac{k}{d^{2}}

01
Set up the model and solve for $k$ using the given point:
$80 = \frac{k}{3^{2}} \implies k = 720$
02
Use $k$ to find the loudness at 6 meters:
$I(6) = \frac{720}{6^{2}} = 20 \text{ dB}$

COMMON MISTAKE

Don't assume every "inversely proportional" situation uses plain $y=k/x$ — always check whether the problem says "inversely proportional to the SQUARE" (or cube, etc.), which changes which product actually stays constant.

Quick check

$I$ is inversely proportional to the square of $d$ . If $d$ doubles, what happens to $I$ ?

Piecewise Functions Algebraically

Worked example

A cyclist's speed $(m/s)$ during a 10-second time trial is modeled by: speed(t) = $3t^{2}$ for 0≤ $t \le 2$ ; speed(t) = 12 for 2< $t \le 6$ ; speed(t) = −2t+24 for 6< $t \le 10$ . Find speed(8), and the average rate of change from $t=0$ to $t=2$ .

01
speed(8): since 6<8≤10, use the third piece: speed(8) = −2(8)+24 = 8 m/s.
02
AROC[0,2]: since both endpoints fall in the first piece, use speed(t)=3t²: (speed(2)−speed(0))/ $(2-0) = (12-0)/2 = 6 m/s^{2}$ .

Practice Problems

Worked example

Problem 1 (Calculator Active). Run a quadratic regression $f(x)$ =ax²+bx+c on this data, then predict $f(2.5)$ .

x	1	2	3	4	5	6
f(x)	12	7	5	5.5	8	13

01
Regression gives $a \approx 1.214$ , $b \approx -8.257$ , $c \approx 18.9$ .
02
$f(2.5) \approx 1.214(6.25) - 8.257(2.5) + 18.9 \approx 5.85$ .

Worked example

Problem 2 (Calculator Active). The data shows monthly rent ($) over years renting.

years	1	3	4	7	9	11	14
rent ($)	1200	1260	1290	1380	1440	1500	1590

01
Linear regression: $S(t) = 30t + 1170$ ( $a=30$ , $b=1170$ — a very clean fit here).
02
Predict rent at year 10: $S(10) = 30(10)+1170$ = $1470.

Worked example

Problem 3. The cost per cookie is inversely proportional to the square root of the batch size. It costs $3/cookie for a batch of 16. Find the cost per cookie for a batch of 36.

01
$k = 3$ ·√16 = 12.
02
Cost at $n=36$ : 12/√36 = $12/6$ = $2 per cookie.

Worked example

Problem 4. A print shop charges $0.10 per page for the first 50 pages: C(p)=0.10p for 0≤p≤50; then $0.06 per page after that: $C(p)=0.06(p-50)+5$ for $p>50$ . Find $C(80)$ .

01
Since 80>50, use the second piece: $C(80) = 0.06(30)+5 = 1.8+5$ = $6.80.

FRQ-Style: Building a Model from 3 Points

Worked example

A Ferris wheel's height above the ground (meters) is recorded at several times (seconds): $(0,3)$ , $(5,25)$ , $(12,40)$ , $(18,38)$ , $(25,20)$ , $(30,5)$ . Using three of these points, set up and solve a system of equations to approximate a quadratic model $h(t)$ =at²+bt+c.

01
Choose three representative points — often the first, one from the middle, and the last: $(0,3)$ , $(12,40)$ , $(30,5)$ .
02
Substitute each into $h(t)$ =at²+bt+c:
03
From $(0,3)$ : $c = 3$ .
04
From $(12,40)$ : 144a+12b+ $c=40$ → 144a+12b=37.
05
From $(30,5)$ : 900a+30b+ $c=5$ → 900a+30b=2.
06
Solve the system (elimination or substitution): $a \approx -0.168$ , $b \approx 5.094$ , $c=3$ .

Residuals: How Good Is the Model?

A regression model won't hit every data point exactly — the residual measures exactly how far off it is at a given point.

KEY RULE

Residual = Actual Value − Predicted Value

Worked example

Using the linear model $S(t) = 30t + 1170$ from Practice Problem 2 (monthly rent), a tenant reports their year-8 rent was $1450 (year 8 wasn't one of the original data points). What is the residual, and what does it mean?

01
Predicted value: $S(8) = 30(8)+1170 = 240+1170$ = $1410.
02
Residual = Actual − Predicted = 1450 − 1410 = $40.
03
Interpretation: the actual year-8 rent was $40 MORE than the model predicted — a positive residual means the actual value sits ABOVE the model's line.

Quick check

A model predicts $41.85$ and the actual value is $43$ . What is the residual, with its sign?

Common slips

Using a different calculator (Casio, a different ti model, an online tool)? The exact button sequence will differ from what's shown here. Search "[your calculator model] + regression" (e.g. "Casio fx-9750 quadratic regression") to find the right steps for your device — the concept (enter data, choose model type, run it) stays the same everywhere.
Regression coefficients from real data are almost never clean integers — round sensibly (usually 3 decimal places) and carry the rounded model forward consistently.

Always double check which regression type matches your data's shape before running it (1.13's tools apply here too).
Don't assume every "inversely proportional" situation uses plain $y=k/x$ — always check whether the problem says "inversely proportional to the square" (or cube, etc.), which changes which product actually stays constant.

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8 cards · Choosing a model

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Recap card

6 lines to re-read the night before.

01
Regression finds the best-fit coefficients for a chosen model type — expect messy decimals, not clean integers, from real data.
02
Recognize the 8 common regression shapes (linear, quadratic, cubic, quartic, exponential, logarithmic, logistic, sine) before choosing one.
03
Inversely proportional: find $k$ from one data point ( $k = x \cdot y$ for the basic form, or $k = x^{2} \cdot y$ for inverse-square), then use $k$ to answer other questions.
04
For piecewise functions, always check which piece's domain condition an x-value satisfies before evaluating.
05
To build a model from just 3 points, substitute each into the general form and solve the resulting system of equations.
06
Residual = Actual − Predicted: positive means the model under-predicted; negative means it over-predicted.

Practice 23 problems Next: Chapter 2.1 Change in Arithmetic and Geometric Sequences

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Function Model Construction

Building a Regression Model on a Calculator (TI-84)

Constructing a Polynomial Model from Data

Types of Regression — a Visual Reference

Inversely Proportional Models

Piecewise Functions Algebraically

Practice Problems

FRQ-Style: Building a Model from 3 Points

Residuals: How Good Is the Model?

Recap card

Unlock Premium

Building a Regression Model on a Calculator (TI-84)

Constructing a Polynomial Model from Data

Types of Regression — a Visual Reference

Inversely Proportional Models

A Related Form: Inverse-SQUARE Proportion

Piecewise Functions Algebraically

Practice Problems

FRQ-Style: Building a Model from 3 Points

Residuals: How Good Is the Model?

Recap card