How do you choose which function type best models a data set?

Look at how the output values change as the input values change, then match that behavior to a known function type. Different function types have characteristic behaviors, which is what makes the match possible.

True or False? For real data, the differences must be exactly constant before a polynomial model of that degree is appropriate.

False. Real data is rarely exact, so differences that are roughly constant are enough to justify the model. That is the main difference between choosing a model and stating an exact property of a known polynomial.

What kind of context suggests a quadratic model, and what suggests a cubic one?

Scenarios involving area , or two dimensions, often call for a quadratic model. Scenarios involving volume , or three dimensions, often call for a cubic one.

What features of a data set suggest a polynomial model?

Multiple real zeros , or multiple maxima or minima , both point to a polynomial. A data set that is roughly symmetric with a single maximum or minimum points specifically to a quadratic.

Define piecewise-defined function .

A piecewise-defined function is a set of functions, each defined over its own non-overlapping domain interval. Different pieces apply to different intervals of the input values, and each piece may be a different type of function.

True or False? A piecewise-defined function is a single function, even though it is built from several different function types.

True. For each input value exactly one of the pieces applies, so every input gives exactly one output. That is precisely what being a function requires, however many pieces are involved.

Why is a piecewise-defined function useful for modeling?

Many real situations show different characteristics over different intervals , which a single function type cannot capture across the whole domain. A piecewise model lets the most appropriate function type be used on each interval.

Do the pieces of a piecewise-defined function have to connect at the boundaries?

No, the output value may jump abruptly from one piece to the next. Whether they connect depends on the scenario: a water level cannot jump instantly, but a pricing structure can change sharply at a threshold.

What are the assumptions underlying a model, and why do they matter?

They are the things taken to be true in order for the model to work. Understanding them is what lets you recognise when a model is appropriate and when it might break down.

What does a linear model for distance travelled assume?

That the speed stays constant throughout the scenario. If the speed changes the model is no longer appropriate, just as a population model assuming a constant growth rate fails once that rate shifts.

What does choosing a quadratic rather than a linear model assume about the rates of change?

Choosing a model commits you to a claim about how the quantities vary together: linear assumes a constant rate of change, quadratic assumes that rate itself changes at a constant rate. If the real relationship changes, for instance if a trend reverses, the model stops being valid beyond that point.

Why might a model need its domain restricted?

Because the function may be defined for all real numbers while the context only makes sense for certain input values. Restrictions come from contextual clues such as time not being negative, from the model's mathematical behavior conflicting with the context, and from the range of the data used to build it.

True or False? A model is equally reliable far outside the range of the data used to build it.

False. A model is generally only reliable within the range of the data it was built from. Predictions far outside that range, which is extrapolation, may be unreliable.

How do you construct a model once you know which function type to use?

Use the given data or conditions to set up equations involving the unknown constants, then solve that system. Those conditions come from specific data values, initial conditions or known behaviors of the quantities involved.

What does a regression do, and what does it need?

A regression uses an algorithm to find the function of a given type that best fits a data set. You choose the type first, whether linear, quadratic, cubic or quartic, then enter the data and let the calculator return the constants.

How do you construct a piecewise-defined model?

Identify the intervals where the behavior changes, choose a function type for each, and build each piece separately by the usual methods. Then combine them, giving a domain for each piece and checking that those intervals do not overlap.

What can a constructed model be used to do?

Predict values by evaluating it at a chosen input, and calculate or approximate rates of change. It can also estimate values from an average rate of change, and describe how the rate of change is itself increasing or decreasing.

What does estimating with an average rate of change assume?

That the function changes at a constant rate equal to that average rate of change. Real functions rarely do, which is why the result is an estimate rather than an exact value.

The graph of a model is concave up on an interval. Is an estimate from the average rate of change too high or too low?

Too high, so it is an overestimate . On a concave up interval the secant line lies above the graph, and the estimate is a point on that secant line.

The graph of a model is concave down on an interval. Is an estimate from the average rate of change too high or too low?

Too low, so it is an underestimate . On a concave down interval the secant line lies below the graph, and the estimate is a point on that secant line.

True or False? An estimate made from an average rate of change lies on the graph of the model.

False. It lies on the secant line joining the two endpoints of the interval, which is generally not the same as the graph. That is exactly why the estimate can come out too high or too low depending on the concavity.

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Modeling with Polynomial & Rational Functions (College Board AP® Precalculus): Flashcards

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How do you choose which function type best models a data set?

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Cards in this collection (37)

How do you choose which function type best models a data set?
Look at how the output values change as the input values change, then match that behavior to a known function type.
Different function types have characteristic behaviors, which is what makes the match possible.
How many data points determine a polynomial model of degree $n$ ?
$n + 1$ points with distinct input values, so $2$ points give a linear model, $3$ a quadratic and $4$ a cubic.
A polynomial of degree $n$ or less can always be found through $n + 1$ points, though a lower degree may happen to fit them too.
True or False?
For real data, the differences must be exactly constant before a polynomial model of that degree is appropriate.
False.
Real data is rarely exact, so differences that are roughly constant are enough to justify the model.
That is the main difference between choosing a model and stating an exact property of a known polynomial.
What kind of context suggests a quadratic model, and what suggests a cubic one?
Scenarios involving area, or two dimensions, often call for a quadratic model.
Scenarios involving volume, or three dimensions, often call for a cubic one.
A function has equally spaced inputs and output values $- 10, - 5, 4, 17, 34$ . Its 1st differences are $5, 9, 13, 17$ . Fill in the two gaps.
The 2nd differences are all $______$ so the data is best modeled by a $______$ function.
The completed sentence is:
The 2nd differences are all 4 so the data is best modeled by a quadratic function.
Two levels of differencing were needed, which is what makes the model degree $2$ .
What features of a data set suggest a polynomial model?
Multiple real zeros, or multiple maxima or minima, both point to a polynomial.
A data set that is roughly symmetric with a single maximum or minimum points specifically to a quadratic.
A data set's 1st and 2nd differences are not constant, but its 3rd differences are. What model should be used?
A cubic model, that is a polynomial of degree $3$ .
The number of levels of differencing needed matches the degree, so three levels means degree $3$ .
Define piecewise-defined function.
A piecewise-defined function is a set of functions, each defined over its own non-overlapping domain interval.
Different pieces apply to different intervals of the input values, and each piece may be a different type of function.
True or False?
A piecewise-defined function is a single function, even though it is built from several different function types.
True.
For each input value exactly one of the pieces applies, so every input gives exactly one output.
That is precisely what being a function requires, however many pieces are involved.
How do you evaluate a piecewise-defined function at a given input?
First decide which interval the input falls in, then use the piece that applies on that interval.
For a function that is $3 x + 1$ when $x < 2$ and $x^{2}$ when $x \geq 2$ , the input $0$ uses the first piece and gives $1$ .
The function $f$ is $3 x + 1$ for $x < 2$ and $x^{2}$ for $x \geq 2$ . Fill in the two gaps.
At $x = 2$ the $______$ piece applies, because that interval uses $\geq$ rather than $>$ , so $f (2) =______$ exactly.
The completed sentence is:
At $x = 2$ the second piece applies, because that interval uses $\geq$ rather than $>$ , so $f (2) = 4$ exactly.
Whether an endpoint uses $\leq$ or $<$ decides which piece owns it, which is why the intervals must never overlap.
Why is a piecewise-defined function useful for modeling?
Many real situations show different characteristics over different intervals, which a single function type cannot capture across the whole domain.
A piecewise model lets the most appropriate function type be used on each interval.
Do the pieces of a piecewise-defined function have to connect at the boundaries?
No, the output value may jump abruptly from one piece to the next.
Whether they connect depends on the scenario: a water level cannot jump instantly, but a pricing structure can change sharply at a threshold.
A parking garage charges a flat rate of $5$ dollars for the first hour, then $3$ dollars for each hour after. What function type suits each piece?
A constant function for the first hour, since the cost does not change over that interval.
A linear function afterwards, since the cost then increases at a constant rate.
What are the assumptions underlying a model, and why do they matter?
They are the things taken to be true in order for the model to work.
Understanding them is what lets you recognise when a model is appropriate and when it might break down.
What does a linear model for distance travelled assume?
That the speed stays constant throughout the scenario.
If the speed changes the model is no longer appropriate, just as a population model assuming a constant growth rate fails once that rate shifts.
What does choosing a quadratic rather than a linear model assume about the rates of change?
Choosing a model commits you to a claim about how the quantities vary together: linear assumes a constant rate of change, quadratic assumes that rate itself changes at a constant rate.
If the real relationship changes, for instance if a trend reverses, the model stops being valid beyond that point.
Why might a model need its domain restricted?
Because the function may be defined for all real numbers while the context only makes sense for certain input values.
Restrictions come from contextual clues such as time not being negative, from the model's mathematical behavior conflicting with the context, and from the range of the data used to build it.
A quadratic models a cumulative total and has a maximum at $t = 8$ . Why does that restrict the domain?
After $t = 8$ the quadratic decreases, but a cumulative total can never decrease.
So the model is only appropriate for $t$ values up to $8$ , which becomes the right-hand boundary of its domain.
True or False?
A model is equally reliable far outside the range of the data used to build it.
False.
A model is generally only reliable within the range of the data it was built from.
Predictions far outside that range, which is extrapolation, may be unreliable.
Fill in the two gaps about restricting the outputs of a model.
A model predicting the number of items sold may need its output $______$ to the nearest integer, and a model predicting a negative distance has output values that should be $______$ altogether.
The completed sentence is:
A model predicting the number of items sold may need its output rounded to the nearest integer, and a model predicting a negative distance has output values that should be excluded altogether.
Range restrictions come from what the output can meaningfully be in the context.
How do you construct a model once you know which function type to use?
Use the given data or conditions to set up equations involving the unknown constants, then solve that system.
Those conditions come from specific data values, initial conditions or known behaviors of the quantities involved.
How do you turn data points into equations for a model's constants?
Substitute each data point into the general form of the model, which gives one equation per point.
For $P (t) = a t^{2} + b t + c$ the point $(0, 12)$ gives $c = 12$ directly, since the other terms vanish.
A population is modeled by $P (t) = a t^{2} + b t + c$ , with $P (0) = 12$ and $P (3) = 15.6$ . Fill in the two gaps.
Substituting $t = 0$ gives $c =______$ and substituting $t = 3$ gives $9 a + 3 b + c =______$ as the second equation.
The completed working is:
Substituting $t = 0$ gives $c = 12$ and substituting $t = 3$ gives $9 a + 3 b + c = 15.6$ as the second equation.
A third data point is needed before all three constants can be pinned down.
Data follows a parabola with vertex at $(3, 5)$ opening upward. How can a model be constructed from that?
Start from the parent function $x^{2}$ and apply the transformations that move its vertex there, giving $f (x) = a {(x - 3)}^{2} + 5$ .
Then use one data point to find the value of $a$ .
What does a regression do, and what does it need?
A regression uses an algorithm to find the function of a given type that best fits a data set.
You choose the type first, whether linear, quadratic, cubic or quartic, then enter the data and let the calculator return the constants.
How do you construct a piecewise-defined model?
Identify the intervals where the behavior changes, choose a function type for each, and build each piece separately by the usual methods.
Then combine them, giving a domain for each piece and checking that those intervals do not overlap.
Define inversely proportional.
Two quantities are inversely proportional when one increases as the other decreases and their product is constant.
If $y$ is inversely proportional to $x$ then $y = \frac{k}{x}$ for some constant $k$ .
How do you construct a model when one quantity varies inversely with the square of another?
Write it in the general form $y = \frac{k}{x^{2}}$ , then substitute a known input-output pair to find $k$ .
If the intensity is $45$ at a distance of $2$ , then $45 = \frac{k}{4}$ gives $k = 180$ and the model $I (d) = \frac{180}{d^{2}}$ .
True or False?
You can check that a data set is inversely proportional by seeing whether the product of the two quantities stays constant.
True.
For $y = \frac{k}{x}$ the product $x y = k$ , which does not change.
So a constant product across the data is a quick confirmation that a rational model of that form is appropriate.
What can a constructed model be used to do?
Predict values by evaluating it at a chosen input, and calculate or approximate rates of change.
It can also estimate values from an average rate of change, and describe how the rate of change is itself increasing or decreasing.
Complete the formula for estimating a function value from a known value and an average rate of change.
$Estimate = f (a) + (average rate of change) \times (c -______)$
The completed formula is:
$Estimate = f (a) + (average rate of change) \times (c - a)$
Here $f (a)$ is the known value at $x = a$ , and $c$ is the input value where the estimate is wanted.
What does estimating with an average rate of change assume?
That the function changes at a constant rate equal to that average rate of change.
Real functions rarely do, which is why the result is an estimate rather than an exact value.
The graph of a model is concave up on an interval. Is an estimate from the average rate of change too high or too low?
Too high, so it is an overestimate.
On a concave up interval the secant line lies above the graph, and the estimate is a point on that secant line.
The graph of a model is concave down on an interval. Is an estimate from the average rate of change too high or too low?
Too low, so it is an underestimate.
On a concave down interval the secant line lies below the graph, and the estimate is a point on that secant line.
A model has $B (0) = 12$ thousand and an average rate of change of $3.5$ thousand per month over $[0, 6]$ . Estimate $B (4)$ .
The estimate is $12 + 3.5 \times 4 = 26$ thousand.
That takes the known value at $t = 0$ and assumes the average rate of change holds all the way across to $t = 4$ .
True or False?
An estimate made from an average rate of change lies on the graph of the model.
False.
It lies on the secant line joining the two endpoints of the interval, which is generally not the same as the graph.
That is exactly why the estimate can come out too high or too low depending on the concavity.

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