# Custom Functions

#### Introduction

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.

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.

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, and a complete set of the online community’s submission is crated on this site under #MenuMath tab.

#### Linear Functions

 Positive y-intercept Negative y-intercept Slope of -2 Positive x-intercept Negative x-intercept Slope of one half Never enters QI Never enters QIV Never enters QIII Goes through the point (-5 , 6) Goes through the origin Never enters QII Slope greater than 3/2 Slope less than 1 Touches both axes at integer coordinates Has a fractional slope value Increases faster than y=-2x + 1 Increases slower than y=1x - 2

*sample of a linear menu (from Amie Albrecht)

 Has a vertex in quadrant I Opens up Opens down Has a vertex in quadrant III Has a maximum value Has a minimum value Has a vertex on the x-axis Positive y-intercept Two x-intercepts Has a vertex on the y-axis Negative y-intercept No x-intercepts Goes through the point (2 , -3) Never enters QII Two negative x-intercepts Horizontally stretched Never enters QIV Two positive x-intercepts Vertically stretched Contains points in all four quadrants Axis of symmetry in Q I & IV

#### Polynomial Functions

 Has 2 turning points Has no turning points Has 2 x-intercepts All x-intercepts are positive Has an even degree Has an odd degree Never enters QIV Never enters QII Has more turning points than x-intercepts Ends in QI Ends in QIV Starts in QII Starts in QIII Goes through (3 , 5) Has a bounded range Always increasing Always increasing in QI Has an infinite range Has more x-intercepts than y-intercepts Has a turning point on the x-axis Has a y-intercept at the origin

#### Exponential Functions

 Is increasing Is decreasing Has a negative x-intercept Has a positive x-intercept Has a negative y-intercept Has a positive y-intercept Has no x-intercepts Has a base > 2 Has 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 value Never enters quadrant I Never enters quadrant II

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

#### Sinusoidal Functions

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

*sample of a sinusoidal menu (from Dylan Kane)

#### Rational Functions

 Has a vertical asymptote in QI & QIV Has a horizontal asymptote in QI & Q2 Has a vertical asymptote in QII & QIII Has 2 vertical asymptotes Numerator has a degree of 3 Has a vertical asymptote in QI & QIV Never touches either axis Has an x-intercept at (4 , 0) Denominator has a degree of 2 Has a positive y-intercept Never enters quadrant III Has 3 x-intercepts The degree of the denominator is greater than the degree of the numerator The degrees of the numerator and denominator are identical Has a negative y-intercept

#### Functions (General)

 Goes through (3 , 4) Horizontal asymptote in QI & QII Negative y-intercept Goes through (-1 , 2) Never enters quadrant I Positive y-intercept Vertical asymptote in QII & QIII Never enters quadrant II Has more x-intercepts than y-intercepts Vertical asymptote in QI & QIV Never enters quadrant III Negative x-intercept Passes through the origin Never enters quadrant IV Positive x-intercept Has a minimum value Has a maximum value Never decreasing Has an infinite range Symmetric about the y-axis Never increasing