The five function families used to model paired data, reading a scatter plot's shape to choose among them, fitting linear, exponential, quadratic and cubic models with regression, and preferring the simpler model when two fit comparably well.
Subject: Algebra 2 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 2 · Chapter 11 — Data Analysis and Statistics
Choose the Best Model for Two-Variable Data
Objectives
Five outcomes. The picture chooses the family; the calculator fits the constants.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-778 — the lesson these objectives are drawn from
Warm-up
Chapters 2, 4, 5, 7 and 8 each introduced a family of functions, and Lesson 7.7 fitted two of them to data.
Discussion prompt
You are handed a table of paired values and asked for a model. Before touching a calculator, what should you do first?
Hint: What does a regression key need you to have decided already?
Answer:
Plot the points. A calculator has a separate regression key for each family, so it cannot choose the family for you — it can only fit the one you ask for.
Every regression will return an equation, including a badly wrong one. The scatter plot is what stops you asking for the wrong family.
So the work of this lesson is reading a shape, and the arithmetic is delegated.
Concept
To model paired data, make a scatter plot and decide from its shape which function family the pattern suggests. Then use the matching regression feature to find the constants.
regression — A calculator feature that finds the constants of a chosen function family giving the best fit to a set of data points. The family must be chosen first, from the shape of the scatter plot.
\[ y = ax+b; \; ax^2+bx+c; \; ab^x; \; ax^b \]
Graphing the fitted model against the data is the final check: a model that follows the points is worth using, and one that does not means the wrong family was chosen.
Figure (svg): The five function families used to model two-variable data, with their general forms
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-777
Section
Section 1
Concept
Linear, quadratic, cubic, exponential and power functions cover the models of this course. Each has its own general form with constants for a regression to determine.
\[ ax+b, \; ax^2+bx+c, \; ax^3+bx^2+cx+d, \; ab^x, \; ax^b \]
The three polynomial families differ only in degree, while the last two put the variable in an exponent or in a base with a constant exponent.
Figure (svg): The five function families used to model two-variable data, with their general forms
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-775 — Function families
Picture it
The five families with their general forms.
Figure (svg): The five function families used to model two-variable data, with their general forms
Each row's shape column is what a scatter plot must match. The forms differ in how many constants a regression has to find, from two up to four.
Worked example
Reading the table of general forms.
\[ \text{Name the family of } y = ab^x, \; y = ax^b, \; y = ax^2+bx+c, \; y = ax+b. \]
First: the variable is the exponent
Why: The base is a constant.
Second: the variable is the base
Why: The exponent is a constant.
Third: degree two
Why: One squared term and two more.
Fourth: degree one
Why: A slope and an intercept.
Figure (svg): The five function families used to model two-variable data, with their general forms
\[ ab^x, \; ax^b, \; ax^2+bx+c, \; ax+b \]
Verify: distinguish the first two carefully
Why: In an exponential the variable sits upstairs and in a power function downstairs, which is Lesson 7.7's distinction exactly. They look similar written down and behave completely differently: an exponential eventually outgrows every power function.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-775
Sorting
Read the general form.
Sort into buckets
Sort each equation.
The last bucket holds two different families, separated by whether the variable is the base or the exponent — 0.969 to the x against x to the 2.5.
Worked example
How much data each family needs.
\[ \text{How many constants does each of the five general forms contain?} \]
Linear and exponential and power
Why: Two letters each, a and b.
\[ 2\text{ constants} \]
Quadratic
Why: Three letters, a, b and c.
\[ 3\text{ constants} \]
Cubic
Why: Four letters, a through d.
\[ 4\text{ constants} \]
Note the consequence
Why: More constants means more flexibility and more data needed.
Figure (svg): The solution to Worked example count the constants shown as a ladder of expressions, one row per algebraic move
\[ 2, \; 3, \; 4, \; 2, \; 2 \]
Verify: connect to Lesson 7.7
Why: Two constants needed two points, and three needed three, which is why a parabola through three points was determined in Lesson 4.10. A cubic with four constants can be made to pass through any four points exactly — which is a warning as much as a capability.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-775
Trap
\[ y = ab^x \text{ and } y = ax^b \]
Treat them as the same family
Why: Both have an a and a b in an exponent-shaped expression.
\[ \text{interchangeable} \quad \text{(wrong)} \]
In the first the variable is the exponent and in the second it is the base. Their graphs and their growth rates are completely different.
\[ ab^x: \; x \text{ upstairs}; \qquad ax^b: \; x \text{ downstairs} \]
Read where the variable sits
Why: That single reading names the family.
\[ 1.42^x \text{ eventually beats } x^{2.5} \]
Lesson 7.7 also gave a data test: exponential data straightens on a semi-log plot and power data on a log-log plot.
Fill the middle
The cubic form.
Fill in the blanks
y = ax^3+bx^2+cx+d \text4 ___ \text___
Why: Four constants, one more than a quadratic and two more than a line. Each extra constant lets the curve bend once more, which is both its power and its risk.
Matching
Five forms, five families.
Match the pairs
Why: The last two differ by a single swap: which of the two symbols is the variable and which is the constant. Everything about their behaviour follows from that swap.
Prediction
Commit before reasoning.
Predict first
A graphing calculator has separate regression keys for each family. What does that tell you?
Correct: You must choose the family first; the calculator only fits the constants of whichever one you pick.
\[ \text{you choose the family; the calculator finds } a, b, c, \dots \]
Why: Every regression key returns an equation, including for a family that fits terribly — a linear regression on a parabola-shaped scatter will produce a line and no warning. So the judgement about which family to use is entirely yours, made from the scatter plot, and it is the only part of the process a machine cannot do for you.
Section
Section 2
Concept
A scatter plot with no turning points suggests a line, one turn suggests a quadratic and two suggest a cubic. A curve with no turn but a changing rate suggests an exponential or a power function.
\[ 0 \text{ turns, } 1 \text{ turn, } 2 \text{ turns} \]
Distinguishing exponential from power data is Lesson 7.7's work: transform the plot and see which version straightens.
Figure (svg): Four scatter plot shapes and the family each one suggests
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-775 — Choose a model from a scatter plot
Picture it
What each pattern of points suggests.
Figure (svg): Four scatter plot shapes and the family each one suggests
The first three are told apart by counting turns. The fourth has no turn at all but a rate that changes steadily, which no polynomial produces so smoothly.
Worked example
Examples 1, 2 and 3, at the plotting stage.
\[ \text{Tuition rises steadily; chili temperature falls fast then levels; fuel efficiency rises then falls. Name each family.} \]
Tuition
Why: The points lie close to a straight line, with no turn.
Cooling
Why: A steep fall flattening toward a floor, with no turn.
Fuel efficiency
Why: An inverted U with one turning point.
Note what distinguished them
Why: Turns, and whether the rate changes.
Figure (svg): Four scatter plot shapes and the family each one suggests
\[ \text{linear}, \; \text{exponential}, \; \text{quadratic} \]
Verify: check the cooling case against a line
Why: A line through the cooling data would keep falling and predict a negative temperature after about 80 minutes. The data flattens instead, approaching the freezer's temperature — which is exactly what an exponential with a floor does and a line cannot.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-777
Sorting
Count the turns.
Sort into buckets
Sort each described scatter plot.
The last is the one a hasty reading turns into a line: it has no turn either, and only the changing rate distinguishes it.
Worked example
Guided Practice 4 and 5.
\[ \text{Data rising to about } 70 \text{ near } x=400 \text{ then falling; and data crossing the axis at } -4, -2 \text{ and } 1. \]
First: count the turns
Why: The values rise to a peak and then fall.
First: name the family
Why: One turn means degree two.
Second: count the crossings
Why: The curve meets the axis three times.
Second: name the family
Why: Three roots need degree at least three.
Figure (svg): The solution to Worked example two more shapes shown as a ladder of expressions, one row per algebraic move
\[ \text{quadratic}; \qquad \text{cubic} \]
Verify: check the second by counting turns too
Why: A curve crossing the axis three times must turn twice between the crossings, and two turns is exactly a cubic. The two readings — counting roots and counting turns — agree, which is a good sign that the shape has been read correctly.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 777-777
Trap
\[ \text{cooling data: } 100, 75, 50, 35, 28, 20, 15 \]
Notice the values falling and fit a line
Why: A steady decrease is taken as evidence of linearity.
\[ y \approx -1.4x+92 \quad \text{(a poor fit)} \]
The first drops are 25 degrees per ten minutes and the last only 5, so the rate is not constant — which is what a line assumes.
\[ y = 98.2(0.969)^x \]
Check whether the rate is constant before choosing a line
Why: Constant differences mean linear; constant ratios mean exponential.
\[ \text{ratios: } 0.75, 0.67, 0.70, 0.80, 0.71, 0.75 \text{ per ten minutes} \]
Computing successive differences and successive ratios takes a minute and settles the choice between the two commonest families.
Comparison
Fill the blanks. Two quick tests on a table.
Comparison matrix
| Test | Constant means | Example |
|---|---|---|
| Successive differences | linear | tuition rising about 933 a year |
| Successive ratios | exponential | chili losing about 3 percent a minute |
| Second differences | quadratic | fuel efficiency |
| Neither | try a power model or a cubic | check a log-log plot |
These arithmetic tests confirm what the picture suggests, and they are worth running whenever two families both look plausible.
Fill the middle
Guided Practice 5.
Fill in the blanks
\text3 ___
Why: A polynomial with three distinct roots has at least three linear factors, so its degree is at least three. Counting axis crossings is a second way to reach the same family as counting turns.
Prediction
Commit before reasoning.
Predict first
Why is making a scatter plot the first step rather than running every regression and picking the best fit?
Correct: Because a higher-degree model always fits at least as well, so best fit alone would always choose the most complex family.
\[ \text{cubic fit} \geq \text{quadratic fit, always} \]
Why: A cubic can reproduce any quadratic exactly by setting its leading coefficient to zero, so it can never fit worse. Choosing purely on fit would therefore always pick the highest degree available, however meaningless. The scatter plot supplies the independent judgement that stops that, which is why the book prefers the simpler model when two fit comparably.
Section
Section 3
Concept
When the points lie close to a straight line, linear regression gives the slope and intercept of the best-fitting line. The slope is the rate of change, in the data's own units.
\[ y = 933x+14{,}600 \]
Rounding the regression output is usual: 933.37 and 14,590.58 become 933 and 14,600, which loses nothing the data could support.
Figure (svg): Tuition data plotted against years, with a fitted straight line
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-775 — Use a linear model
Picture it
Example 1: eight years of average private college tuition.
Figure (svg): Tuition data plotted against years, with a fitted straight line
The line follows the points closely, and its slope of about 933 says tuition rose by roughly 933 dollars a year over the period.
Worked example
Example 1.
\[ \text{Tuition from } 14{,}537 \text{ to } 21{,}183 \text{ over } 8 \text{ years. Find a model.} \]
Make a scatter plot
Why: The points lie approximately on a line.
Run linear regression
Why: The calculator returns a and b.
\[ 933.37\text{ and } 14, 590.58 \]
Round sensibly
Why: The data has no more precision than this.
\[ y = 933 x + 14, 600 \]
Graph the model with the data
Why: The line passes among the points.
Figure (svg): Tuition data plotted against years, with a fitted straight line
\[ y = 933x+14{,}600 \]
Verify: check the model at both ends
Why: At x equal to 0 the model gives 14,600 against an actual 14,537, and at x equal to 7 it gives 21,131 against 21,183 — both within about 60 dollars on figures near 20,000. A model that tracks the data at both ends and in the middle is doing its job.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-775
Fill the middle
Example 1.
Fill in the blanks
\text933 ___ \text___
Why: The slope of 933 is the yearly increase in dollars, since x counts years and y counts dollars. Reading a slope in the data's units is what turns a fitted constant into an answer.
Worked example
Guided Practice 2.
\[ \text{For } x = 0 \text{ to } 7 \text{ with } y = 33, 41, 52, 68, 80, 89, 102, 118, \text{ find a model.} \]
Check the differences
Why: Eight, 11, 16, 12, 9, 13 and 16.
Plot the points
Why: They rise almost in a straight line.
Run the regression
Why: The slope and intercept.
\[ \text{about } 12.2\text{ and } 30.3 \]
Write the model
Why: Rounded to the data's precision.
\[ y = 12.2 x + 30.3 \]
Figure (svg): The solution to Worked example a second linear fit shown as a ladder of expressions, one row per algebraic move
\[ y \approx 12.2x+30.3 \]
Verify: check the total rise
Why: The values climb from 33 to 118 over seven steps, an average of about 12.1 per step — matching the fitted slope of 12.2. Averaging the overall change is a quick estimate of any linear slope and a good check on a regression.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 776-776
Error analysis
A student reports the tuition model straight from the calculator.
Annotate
On: \( y = 933.3690476x+14590.58333 \)
A model should be reported to about the precision of the data it came from. Extra digits are noise dressed up as information.
Sorting
For y equals 933x plus 14,600.
Sort into buckets
Sort each interpretation.
The intercept has a real meaning here because x equal to 0 is a real year, 1995. In many models it does not, and reading it as a fact about the world would be a mistake.
Matching
Constant differences mean linear.
Match the pairs
Why: Both models have the same structure and completely different constants, which is why the family and the fit are separate steps. The family says what kind of answer to expect; the regression says which one.
Prediction
Commit before reasoning.
Predict first
The tuition model covers 1995 to 2002. What would it predict for 2050, and should you believe it?
Correct: About 65,000 dollars, but no — the model is fitted to eight years and nothing guarantees the trend continues.
\[ 933(55)+14{,}600 \approx 65{,}900 \]
Why: Substituting x equal to 55 gives 933 times 55 plus 14,600, about 65,900. The arithmetic is easy and the extrapolation is not justified: economic conditions, policy and inflation over fifty years are nowhere in the data. This is the same extrapolation warning as Lesson 7.7's scooter model, and it applies to every fitted model regardless of how well it matches the points it was built from.
Section
Section 4
Concept
Data that falls steeply and then levels off, or rises ever faster, suggests an exponential model. Exponential regression finds the initial value and the growth or decay factor.
\[ y = 98.2(0.969)^x \]
A decay factor of 0.969 means about 3.1 percent of what remains is lost each minute, which is Lesson 7.2's reading of the base applied to fitted data.
Figure (svg): Cooling temperatures plotted against time, with a fitted exponential decay curve
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 776-776 — Use an exponential model
Picture it
Example 2: chili temperature in a freezer.
Figure (svg): Cooling temperatures plotted against time, with a fitted exponential decay curve
The curve drops fast at first and flattens toward the bottom, which is exactly what a line cannot do and an exponential does naturally.
Worked example
Example 2.
\[ \text{Temperatures } 100, 75, 50, 35, 28, 20, 15 \text{ at } 0 \text{ to } 60 \text{ minutes. Find a model.} \]
Make a scatter plot
Why: The points fall rapidly and then level off.
Run exponential regression
Why: The calculator returns a and b.
\[ 98.24\text{ and } 0.9687 \]
Round sensibly
Why: Three significant figures suffice.
\[ y = 98.2(0.969) ^{x} \]
Graph the model with the data
Why: The curve follows the points.
Figure (svg): Cooling temperatures plotted against time, with a fitted exponential decay curve
\[ y = 98.2(0.969)^x \]
Verify: check the model at two times
Why: At x equal to 0 the model gives 98.2 against an actual 100, and at x equal to 30 it gives 98.2 times 0.969 to the thirtieth, about 38.6, against an actual 35. The fit is close without being exact, which is what real data produces.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 776-776
Fill the middle
Example 2.
Fill in the blanks
1-0.969 = 0.031 \text___
Why: About 3.1 percent of the remaining heat is lost each minute. A base below 1 always means decay, and one minus the base gives the rate.
Worked example
Reading the fitted constants.
\[ \text{What do } 98.2 \text{ and } 0.969 \text{ mean in the cooling model?} \]
Read a
Why: The value when x is 0.
\[ \text{about } 98 ^\circ\text{ at the start} \]
Read b
Why: The factor by which y multiplies each minute.
\[ 0.969 \]
Convert b to a percent
Why: One minus 0.969.
\[ 3.1 \%\text{ lost per minute} \]
Note the shape
Why: A base below 1 means decay.
Figure (svg): The solution to Worked example interpret the base shown as a ladder of expressions, one row per algebraic move
\[ a = 98.2; \quad 1-0.969 = 3.1\% \]
Verify: check the ten-minute factor
Why: Zero point nine six nine to the tenth is about 0.73, so roughly 27 percent should be lost every ten minutes. The data falls from 100 to 75 in the first ten minutes, a loss of 25 percent — close, and confirming that the per-minute factor is being read correctly.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 776-776
Trap
\[ y = 98.2(0.969)^x \text{ at } x = 300 \]
Predict the temperature after five hours
Why: The model is applied outside its data range.
\[ y \approx 0.0075\degree F \quad \text{(impossible)} \]
The chili cannot fall below the freezer's temperature, but the model heads to zero and beyond into meaninglessness.
\[ \text{the data covers } 0 \text{ to } 60 \text{ minutes} \]
Use the model only near the range it was fitted to
Why: Cooling levels off at the surrounding temperature, which this model has no way to represent.
\[ \text{a full model would be } y = (T_0-T_R)e^{-rt}+T_R \]
Lesson 7.6's Newton's law of cooling includes the surrounding temperature, which is exactly the floor the fitted exponential lacks.
Sorting
Compare the base with 1.
Sort into buckets
Sort each exponential model.
The base is doing all the work in every case, and comparing it with 1 is a one-glance reading of the model's behaviour.
Comparison
Fill the blanks. Two very different assumptions.
Comparison matrix
| Question | Linear model | Exponential model |
|---|---|---|
| What stays constant | the difference per step | the ratio per step |
| Cooling data | a poor fit; the rate is not constant | a good fit |
| Long-run behaviour | keeps falling past zero | approaches zero and never crosses |
| Which the data supports | neither past 60 minutes | the exponential, within the data range |
Neither model should be pushed far past the data, but only one of them fits the data it was built from.
Prediction
Commit before reasoning.
Predict first
The chili loses 25 degrees in the first ten minutes and 5 in the last ten. What does that pattern rule out?
Correct: A linear model, since a line assumes a constant loss per minute.
\[ \text{losses: } 25, 25, 15, 7, 8, 5 \text{ per ten minutes} \]
Why: A line's defining property is a constant rate of change, and here the rate falls fivefold across the hour. What is roughly constant is the RATIO: about three quarters of the temperature above the surroundings survives each ten minutes. Constant differences mean linear and constant ratios mean exponential, which is the arithmetic version of reading the scatter plot's shape.
Section
Section 5
Concept
One turning point suggests a quadratic and two suggest a cubic. When both fit a data set well, the simpler model is usually the better choice.
\[ y = -0.00793x^2+0.727x+13.8 \]
A cubic can always match a quadratic's fit, since setting its leading coefficient to zero reproduces one exactly — so a better fit alone is never a reason to prefer it.
Figure (svg): Fuel efficiency plotted against speed, with a fitted parabola
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 777-777 — Use a quadratic model
Picture it
Example 3: fuel efficiency against speed.
Figure (svg): Fuel efficiency plotted against speed, with a fitted parabola
Efficiency rises to a peak near 46 miles per hour and falls away, which is one turning point and therefore a quadratic.
Worked example
Example 3 and Guided Practice 3.
\[ \text{Efficiency peaks around } 30 \text{ mpg near } 50 \text{ mph. Find a model and predict the value at } 70 \text{ mph.} \]
Make a scatter plot
Why: The points form an inverted U.
Run quadratic regression
Why: The calculator returns a, b and c.
\[ -0.00793, 0.727, 13.8 \]
Write the model
Why: In standard form.
\[ y = -0.00793 x ^{2} + 0.727 x + 13.8 \]
Substitute 70
Why: Negative 38.9 plus 50.9 plus 13.8.
\[ \text{about } 25.8\text{ mpg} \]
Figure (svg): Fuel efficiency plotted against speed, with a fitted parabola
\[ y(70) \approx 25.8 \text{ mpg} \]
Verify: locate the vertex
Why: The vertex sits at negative b over 2a, which is 0.727 over 0.01586, about 45.8 miles per hour — the most economical speed. The data's highest value is at 55 mph, close enough given the scatter, and the model's peak is the study's actual answer.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 777-777
Fill the middle
Example 3.
Fill in the blanks
x = -\frac45.8___ = \frac______ \approx ___ \text___
Why: The vertex sits near 45.8 miles per hour, the most economical speed. Reading a quadratic model's vertex is usually the whole point of fitting one.
Worked example
Guided Practice 5.
\[ \text{Data at } x = -5,-4,-3,-2,-1,1,2 \text{ gives } y = -20, 0, 3, 0, -4, 0, 18. \text{ Choose a family.} \]
Find the axis crossings
Why: Y is zero at x equal to negative 4, negative 2 and 1.
Count the turns
Why: The curve must turn between each pair of roots.
Name the family
Why: Two turns means degree three.
Write an approximate model
Why: Roughly 0.75 times the product of the three factors.
\[ y\text{ about } 0.75(x + 4) (x + 2) (x - 1) \]
Figure (svg): The solution to Worked example recognise a cubic shown as a ladder of expressions, one row per algebraic move
\[ y \approx 0.75(x+4)(x+2)(x-1) \]
Verify: test the factored form at two points
Why: At x equal to 2 the model gives 0.75 times 6 times 4 times 1, which is 18 — matching exactly. At x equal to negative 3 it gives 0.75 times 1 times negative 1 times negative 4, which is 3, also matching. Three roots read straight off a table is the fastest route to a cubic's shape.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 777-777
Error analysis
A student compares two regressions on the fuel efficiency data.
Annotate
On: \( \text{the cubic fits better, so use the cubic} \)
The book's own note says it: when both fit, choose the simpler model. A better fit that was inevitable carries no information.
Sorting
Turns decide the degree.
Sort into buckets
Sort each pattern.
A degree-n polynomial has at most n minus 1 turns and at most n roots, so either count gives a lower bound on the degree needed.
Two truths and a lie
All three are about choosing between models.
Eliminate the wrong options
Two of these are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false. Fit alone always rewards complexity, so a rule of best fit would always choose the highest degree available — and a high-degree curve bending to follow random scatter predicts badly outside the data. The scatter plot's shape, not the fit statistic, is what should choose the family.
Prediction
Commit before reasoning.
Predict first
A quadratic and a cubic both fit a data set well. Why choose the quadratic?
Correct: Because the extra constant lets a cubic bend to follow random scatter, which predicts worse outside the data.
\[ \text{more constants} \;\Longrightarrow\; \text{more flexibility} \;\Longrightarrow\; \text{more noise fitted} \]
Why: Every data set contains measurement noise, and a more flexible curve fits that noise as eagerly as it fits the real pattern. The result looks better on the points it was built from and worse everywhere else. Choosing the simplest family the shape supports is a guard against mistaking noise for signal, and it also gives a model whose constants can be interpreted — a quadratic's vertex means something, and a cubic's extra wiggle usually does not.
Comparison
Fill the blanks. The picture chooses.
Comparison matrix
| Scatter plot shape | Family | Example from the lesson |
|---|---|---|
| A straight line | linear | tuition rising 933 a year |
| One turning point | quadratic | fuel efficiency against speed |
| Two turning points | cubic | data with three x-intercepts |
| Steep then levelling | exponential | chili cooling in a freezer |
Every row is a judgement about a picture, made before any regression key is pressed.
Pattern
Plot, choose, fit, check.
A higher-degree model always fits at least as well, so a better fit is never on its own a reason to prefer one.
OpenStax Algebra and Trigonometry 2e, §4.3 Fitting Linear Models to Data §4.3
Check
Read the shape.
Check your understanding
Data falls steeply at first and then levels off toward a floor. Which family fits?
Answer: A
Why: A falling rate with no turning point is exponential decay.
Check
Interpret the slope.
Check your understanding
Tuition data gives y = 933x + 14,600 with x in years since 1995. What does 933 mean?
Answer: A
Why: The slope is the change in y per unit increase in x, which is dollars per year.
Check
Simpler is usually better.
Check your understanding
A quadratic and a cubic both fit a data set well. Which should you choose?
Answer: A
Why: A cubic always fits at least as well, so the better fit carries no information.
Real world
In early 2020 the daily count of confirmed cases of a new disease grew from 1 to 1000 over about ten weeks, roughly multiplying by 2 every five days.
Discussion prompt
Decide which family models that growth, project the count ten weeks further, and say why the projection cannot be right.
Hint: A constant doubling time means a constant ratio.
Answer:
\[ \text{doubling every } 5 \text{ days} \;\Longrightarrow\; y = ab^x, \; b = 2^{1/5} \approx 1.149 \]
\[ \text{ten more weeks} = 14 \text{ more doublings} \;\Longrightarrow\; 1000 \times 2^{14} \approx 16{,}000{,}000 \]
The model is exponential, and extended ten weeks it predicts sixteen million daily cases — more than most countries could ever produce.
Exponential growth is always temporary, because it eventually runs out of whatever it is consuming: susceptible people, food, market, fuel. Real epidemic curves bend over into an S shape as the supply of people to infect runs down, and the standard model for that is a logistic curve rather than any of this lesson's five families. The lesson's own warning is what applies: a model fitted over one range describes that range, and extrapolating far past it is where confident forecasts go wrong. Reading the early data as exponential was correct; believing the extrapolation would not have been.
Commit first
Answer, then rate your confidence honestly.
Predict first
A cubic regression fits your data better than a quadratic. Is that a reason to use the cubic?
Correct: No — a cubic can always match a quadratic, so a better fit is guaranteed and carries no information.
\[ a = 0 \;\Longrightarrow\; ax^3+bx^2+cx+d = bx^2+cx+d \]
Why: Setting a cubic's leading coefficient to zero reproduces any quadratic exactly, so the cubic's best fit can never be worse. Choosing on fit alone would therefore always select the highest degree available, however meaningless — and a flexible curve bending to follow random scatter predicts badly outside the data it was built from. The scatter plot's shape is the independent evidence that should decide, and the book states the rule directly: when both fit, prefer the simpler model.
Explain it
They know how to use a regression key and reach for whichever one is nearest.
Discussion prompt
In four sentences or fewer, explain how to decide which regression to use.
Hint: Look at the picture first.
Answer:
Plot the points before you touch a regression key. If they lie on a straight line, use linear; if they rise to a peak and fall, use quadratic; if they fall fast and then flatten out, use exponential.
The calculator will happily fit any family you ask for, including a badly wrong one, so choosing the family is your job. Then graph the fitted model on top of the points and check that it actually follows them.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For the first, ask whether the variable is the base or the exponent. For shapes, count the turning points. For interpretation, attach the data's units to each constant and read it as a sentence. For deciding, take the simplest family the shape supports rather than the best fit.
Connect it up
Paper. Fifteen minutes.
Draw it
Build a modelling page. Top: list the five families with their general forms, how many constants each has, and a small sketch of the shape each produces. Middle left: plot the tuition data, draw the fitted line, and write one sentence interpreting both constants in dollars and years. Middle right: plot the cooling data, sketch the fitted curve, and write what a line would predict after 80 minutes and why that is impossible. Bottom left: plot the fuel efficiency data, sketch the parabola, compute the vertex, and state the most economical speed. Bottom right: write the rule about preferring the simpler model, and beside it one sentence explaining why a cubic can never fit worse than a quadratic.
If your cooling sketch reaches the horizontal axis, redo it: an exponential decay curve approaches zero without ever touching it.
Recap
Five things, and the first one is looking rather than computing.
| If you see | Then |
|---|---|
| A straight-line pattern | Linear regression |
| One turning point | Quadratic regression |
| Two turning points | Cubic regression |
| A steep fall levelling off | Exponential regression |
| The variable as an exponent | y = ab^x; as a base, y = ax^b |
| Two models fitting equally well | Choose the simpler |
That closes Chapter 11. Chapter 12 turns to sequences and series, where the terms follow a rule and their sums have closed formulas.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.5 Choose the Best Model for Two-Variable Data §11.5, pp. 775-778 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 2 — $55/session, free consultation.