## Functions

This course is based on the following Essential Questions: How can we use functions to make predictions about the world? Should judgments be based on quantified relationships?

Students will learn about and analyze three types of functions: Linear, Quadratic, and Exponential

Significant Assignments:

In this mastery project, you will investigate which kind of function best models your data.

Formulate:
1.Give an in depth description your situation, including any background information and images that make your investigation as clear as possible.
3.Explain why it makes sense for your data set to be modeled by that function.

Represent:
4.Show and explain in as much detail as possible:
a.Your variables: what do “x” and “y” stand for?
b.Tables: data table and a table that models your data
c.Graph: clearly indicate your scale and units
d.Equation: identify each of the coefficients
5.In your opinion, which representation (table, graph, or equation) best reveals the nature of your function or gives you the most information about your function?

Analyze:
6.Show and explain interpolated and extrapolated predictions for your function.
7.Choose and explain a domain and a range for your function.
8.Explain the meanings of your coefficients: Pick any coefficient. If it were to change, how would your function be affected? Do the same for all coefficients.
9.How do your table, equation and graph help people understand the problem or situation your data describes?
10.Analyze the graph – what do all the intercepts, extreme points, or important points mean in the context of the situation?
11.Each student gets one extension question from their teacher.

Interpret:
12.During a class lesson planned by your teacher, you will make sense about at least one other person’s problem.

Significant Activities or Projects:

Identify patterns in a variety of forms (stories, geometric patterns, linear patterns).
Represent patterns (xy-relationshps) using words, tables, graphs and equations.
Describe growth that either increases or decreases by a constant amount.
Write an equation of a linear function in slope-intercept form.
Identify the slope and the y-intercept from the equation of linear function.
Use a variety of situations to collect and organize data (Barbie bungee, counting beats, balloon drop).
Use linear regression to model real-world data.
Use a linear regression to make predictions about linearly modeled situations.
Given a situation that can be modeled using a step function (postage rates, etc…), create a graph to model the situation.
Recognize non-integer values within an interval of values.
Use inequalities to describe intervals.
Come up with situations that can be modeled using step functions.

Identify more than one way of describing the “change” in the Step Change pattern.
Identify one way of describing the change as linear and another way as “non-linear.”
Describe the nature of the change of the non-linear.
In their own words, explain the idea that the quadratic growth means that the difference is linear.
Develop a strategy for the “chip-swap” game.
Using the standard form of the quadratic equation, identify how far a runner is after various amounts of time running.
Plug in x-values to find y-values in quadratic functions.
Find out how far the runner is after further amounts of time by following the pattern (similar to Step Change and Chip Swap).
Identify the standard form of a quadratic equation.
Make up quadratic equations in standard form.
Identify the a, b and c-coeffieients of quadratic equations in standard form.
Develop rules for the effect of the a- and b-coefficients on the graph of the equation.
Apply –b/2a as the x-value of the turning point on the graph.
Given an equation for a quadratic equation in standard form, shift the parabola up/down, right/left, make it skinnier/wider, and flip it.
Determine an object’s height off the ground after various seconds using an equation (with factors for speed, initial height off the ground, and gravitational pull back towards the earth).
Given an object’s initial speed, initial height off the ground, and gravitational pull back towards the earth, write an equation for the object’s position as a function of time.
Determine the amount of time it will take an object to rise and then land in a well below the surface of the earth.
Given data for the relationship between time and an object’s position off the ground, write a quadratic equation.
Create a reasonable pricing system for pizzas based on square units of area in the pizza and another based on diameter.
Discuss the differences in the relationships between “size” and “price.”
Given a specific amount of fencing, sketch various rectangles with perimeter equal to the amount of fence. (for various amounts of fencing).
Determine the dimensions that provide the largest area for each.
Create a generalization for the dimensions with largest area with perimeter P.
Create a rulebook for the rules of exponents.
Identify the growth pattern of exponential functions.

Sample PBATs:
Coffee Cooling: How Long is Too Long?
Additional PBAT questions grow out of the activities listed above