### Clarifications

*Clarification 1*: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; end behavior and asymptotes.

*Clarification 2*: Within the Algebra 1 course, functions other than linear, quadratic or exponential must be represented graphically.

*Clarification 3*: Within the Algebra 1 course, instruction includes verifying that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Functions

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Domain
- Intercept
- Range
- Slope

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students interpreted the slope and $y$-intercept of a linear equation in two variables. In Algebra I, students compare key features of two or more linear or nonlinear functions. Except for quadratic and exponential functions, nonlinear functions must be represented graphically. In later courses, students will compare key features of nonlinear functions represented graphically, algebraically, or with written descriptions.- Within this benchmark, one of the functions given must be linear and the number of functions being compared is not limited to two.
- Problem types include comparing linear to nonlinear functions represented graphically and also opportunities that present linear, quadratic and exponential functions in different forms.
- Instruction includes student exploration of linear, quadratic, and exponential models to
ultimately determine that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly or quadratically.
- For example, provide the following context.
- You are being contracted by a large company to provide technical services to a major engineering project. The contract will involve you advising a group of engineers for the three weeks. The company offers you a choice of two methods of payment for your services. The first is to receive $500 per day of work. The second is to receive payment on a scale: two cents for one total day of work, four cents for two total days, eight cents for three total days, etc. Which method of payment would you choose?

- Have students choose a method of payment and begin a class discussion regarding
the reasoning students used to make their choices
*(MTR.4.1).*Ask if students can create a function to represent each payment method*(MTR.7.1).*

- For example, provide the following context.
- Instruction includes representing domain and range using words, inequality notation and
set-builder notation.
- Words
- If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number.”

- Inequality notation
- If the domain is all values of $x$ greater than 2, it can be represented as $x$ > 2.

- Set-builder notation
- If the domain is all values of $x$ less than or equal to zero, it can be represented as {$x$|$x$ ≤ 0} and is read as “all values of $x$ such that $x$ is less than or equal to zero.”

- Words

### Common Misconceptions or Errors

- When describing domain or range, students may assign their constraints to the incorrect variable. In these cases, ask reflective questions to help students examine the meaning of the domain and range in the problem.
- Students may also miss the need for compound inequalities when describing domain or range. In these cases, use a graph of the function to point out areas of their constraint that would not make sense in context.
- When describing intervals where functions are increasing, decreasing, positive or negative, students may represent their interval using the incorrect variable.

### Strategies to Support Tiered Instruction

- Where students are struggling with concepts such as when a function is increasing,
decreasing, positive, negative or questions about its end behavior, ask reflective
questions:
- Imagine walking on the graph from left to right, where would you be going uphill (increasing) or where would you be going downhill (decreasing)?
- On the left, where are you coming from (far below or far above), and on the right, would you eventually be going up forever or down forever (end behavior)?

### Instructional Tasks

*Instructional Task 1 (*

*MTR.7.1*)- Nancy works for a company that offers two types of savings plans. Plan A is represented by
the function $g$($x$) = 250 + 3$x$, where $x$ is the number of quarter years she has utilized the
plan. Plan B is represented by the function ($x$) = 250(1.01)
^{$x$}, where $x$ is the number of quarter years she has utilized the plan.- Part A. Nancy wants to have the highest savings possible after five years, when she plans to leave the company. Which plan should she use?
- Part B. What if Nancy stays for ten years? Which plan should she use?

Instructional Task 2 (MTR.3.1)

Instructional Task 2 (MTR.3.1)

- Three functions are represented below, with the table representing a linear function. Which function has the smallest $x$-intercept?

### Instructional Items

*Instructional Item 1*

- The functions $f$($x$) and $g$($x$) are shown below, with $g$($x$) representing a linear function. Which function has the greater $y$-intercept?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Perspectives Video: Experts

## Problem-Solving Task

## MFAS Formative Assessments

Students are asked to compare a linear function and an exponential function in context.

Students are given a linear function represented by an equation and an exponential function represented by a graph in a real-world context and are asked to compare the rates of change of the two functions.

Students are asked to provide an example of a nonlinear function and explain why it is nonlinear.

Students are asked to interpret key features of a graph (intercepts and intervals over which the graph is increasing) in the context of a problem situation.

## Student Resources

## Problem-Solving Task

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Task

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task