Custom Functions

Introduction

This page is dedicated to the conjecture that modelling work doesn’t need to take place in a context. What’s important is the direction of the work: that it asks students to build a model to specifications before turning around and employing the model to solve for unknowns. The activity of designing custom models to fit specific constraints is what I term “Upstream Thinking,” and this design process fits quite nicely in our algebraic-minded courses if we use task structures that encourage students to make a high number of design decisions and analyze the fallout from said decisions. Here, you will find lists of constraints that pertain to specific types of functions you and your class may be studying. They are the building blocks for three types of contextless modelling tasks that I have used with my students: boxes tasks, wanted tasks, and menu tasks. Each type of task is described below. The hope is you use the lists of constraints to design tasks of these types to encourage students to build custom models to specifications–to think upstream. (These lists are a starting point; please get creative and feel free to pass along any inventions!)

Boxes Task

A boxes task takes the parameters of the equation of a function and replaces them with empty boxes. The students are then asked to fill in the boxes with values to satisfy a set of constraints (Typically 3-4 constraints are given to the learners all at once). The numbers available for entry into boxes can be restricted to sponsor further thinking. (This type of problem has become popularized by the work of openmiddle.com).

Using the integers between 0 and 9 (inclusive), fill in the boxes so that this sinusoidal function never enters quadrant IV, has a period length shorter than 360 degrees, and has a minimum value at its y-intercept.

Wanted Task

A wanted task (name inspired by Chris Hunter) asks students to build and shift the design of a model as a list of constraints is released one at a time. After each constraint, students need to verify that their current model satisfies the new constraint, or alter their function to meet this new requirement. The model is appropriate if it satisfies every constraint. Teachers can pre-sequence a list of constraints or choose them randomly–but this makes it more likely that a model satisfying each constraint is impossible to build. This is an important conversation, but impossibility is not the ideal result for every wanted task. Examples of wanted tasks asking for quadratic models can be found in this post.

Menu Task

A menu task gives students a longer list of constraints (around 8-10) and asks them to build as many functions as necessary so that each constraint is satisfied with at least one of their custom functions. There is no restrictions for the values of the parameters, and all the menu’s constraints are given at once. Rich opportunity arises as students attempt to resolve the menu with as few functions as possible. They begin to analyze which constraints pair well together and which cannot pair together. A more detailed rationale can be read in this post.

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Linear Functions

Positive y-interceptNegative y-interceptSlope of -2
Positive x-interceptNegative x-interceptSlope of one half
Never enters QINever enters QIVNever enters QIII
Goes through the point (-5 , 6)Goes through the originNever enters QII
Slope greater than 3/2Slope less than 1Touches both axes at integer coordinates
Has a fractional slope valueIncreases faster than y=-2x + 1Increases slower than y=1x - 2

*sample of a linear menu (from Amie Albrecht)

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Quadratic Functions

Has a vertex in quadrant IOpens upOpens down
Has a vertex in quadrant IIIHas a maximum valueHas a minimum value
Has a vertex on the x-axisPositive y-interceptTwo x-intercepts
Has a vertex on the y-axisNegative y-interceptNo x-intercepts
Goes through the point (2 , -3)Never enters QIITwo negative x-intercepts
Horizontally stretchedNever enters QIVTwo positive x-intercepts
Vertically stretchedContains points in all four quadrantsAxis of symmetry in Q I & IV

*sample of a quadratic menu (from yours truly)

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Polynomial Functions

Has 2 turning pointsHas no turning pointsHas 2 x-intercepts
All x-intercepts are positiveHas an even degreeHas an odd degree
Never enters QIVNever enters QIIHas more turning points than x-intercepts
Ends in QIEnds in QIVStarts in QII
Starts in QIIIGoes through (3 , 5)Has a bounded range
Always increasingAlways increasing in QIHas an infinite range
Has more x-intercepts than y-interceptsHas a turning point on the x-axisHas a y-intercept at the origin

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Exponential Functions

Is increasingIs decreasingHas a negative x-intercept
Has a positive x-interceptHas a negative y-interceptHas a positive y-intercept
Has no x-interceptsHas a base > 2Has a base < 1
Goes through the point (2 , 3)Goes through the point (2, 1/2)Goes through the origin
Intercepts the y-axis at a fractional valueNever enters quadrant INever enters quadrant II

*sample of an exponential menu (from Sheri Walker & Tania Asselstine)

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Sinusoidal Functions

Has an amplitude of 3Has a midline above the x-axisHas a midline below the x-axis
Never enters QIIINever enters QIIHas a period length greater than 2π
Has a y-intercept at a maximum valueHas a y-intercept at a minimum valueHas a period length less than π
Is non-symmetric about the x-axisIs symmetric about the y-axisHas a max value at (π/2, 4)
Is phase shifted to the leftGoes through the point (2π, 0)Is phase shifted more than one period length

*sample of a sinusoidal menu (from Dylan Kane)

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Rational Functions

Has a vertical asymptote in QI & QIVHas a horizontal asymptote in QI & Q2Has a vertical asymptote in QII & QIII
Has 2 vertical asymptotesNumerator has a degree of 3Has a vertical asymptote in QI & QIV
Never touches either axisHas an x-intercept at (4 , 0)Denominator has a degree of 2
Has a positive y-interceptNever enters quadrant IIIHas 3 x-intercepts
The degree of the denominator is greater than the degree of the numeratorThe degrees of the numerator and denominator are identicalHas a negative y-intercept

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Functions (General)

Goes through (3 , 4)Horizontal asymptote in QI & QIINegative y-intercept
Goes through (-1 , 2)Never enters quadrant IPositive y-intercept
Vertical asymptote in QII & QIIINever enters quadrant IIHas more x-intercepts than y-intercepts
Vertical asymptote in QI & QIVNever enters quadrant IIINegative x-intercept
Passes through the originNever enters quadrant IVPositive x-intercept
Has a minimum valueHas a maximum valueNever decreasing
Has an infinite rangeSymmetric about the y-axisNever increasing

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Calculus

Has a relative maximum in QIHas 3 critical pointsHas a vertical asymptote
Has a single x-interceptIs not differentiable across its domainIs not continuous across its domain
Has a positive y-interceptHas a negative y-interceptIs always decreasing in quadrant II
Never enters QIHas at least two points of inflectionHas an inflection point in QIV
Has a local minimum in QIIGoes through the point (3, 5)Its y-intercept is a critical point

*sample of a calculus functions menu (from Erick Lee)

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