### Examples

*Algebra 1 Example:*The function models Alicia’s position in miles relative to a water stand

*x*minutes into a marathon. Evaluate and interpret for a quarter of an hour into the race.

### Clarifications

*Clarification 1*: Problems include simple functions in two-variables, such as f(x,y)=3x-2y.

*Clarification 2*: Within the Algebra 1 course, functions are limited to one-variable such as f(x)=3x.

**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

- Function Notation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In middle grades, students worked with $x$-$y$ notation and substituted values in expressions and equations. In Algebra I, students work with $x$-$y$ notation and function notation throughout instruction of linear, quadratic, exponential and absolute value functions. In later courses, students will continue to use function notation with other function types and perform operations that combine functions, including compositions of functions.- Instruction leads students to understand that $f$($x$) reads as “$f$ of $x$” and represents an output of a function equivalent to that of the variable $y$ in $x$-$y$ notation.
- Instruction includes a series of functions with random inputs so that students can see the
pattern that emerges
*(MTR.5.1).*- For example,

- Students should discover that the number in parenthesis corresponds to the input or $x$-value on the graph and the number to the right of the equal sign corresponds to the output or $y$-value.
- Although not conventional, instruction includes using function notation flexibly.
- For example, function notation can been seen as $h$($x$) = 4$x$ + 7 or 4$x$ + 7 = $h$($x$).

- Instruction leads students to consider the practicality that function notation presents to
mathematicians. In several contexts, multiple functions can exist that we want to consider
simultaneously. If each of these functions is written in $x$-$y$ notation, it can lead to
confusion in discussions.
- For example, the equations $y$ = −2$x$ + 4 and $y$ = 3$x$ + 7. Representing these functions in function notation allows mathematicians to distinguish them from each other more easily (i.e., $f$($x$) = −2$x$ + 4 and $g$($x$) = 3$x$ + 7).

- Function notation also allows for the use of different symbols for the variables, which can
add meaning to the function.
- For example, $h$($t$) = −16$t$
^{2}+ 49$t$ + 4 could be used to represent the height, $h$, of a ball in feet over time, $t$, in seconds.

- For example, $h$($t$) = −16$t$
- Function notation allows mathematicians to express the output and input of a function
simultaneously.
- For example, $h$(3) = 7 would represent a ball 7 feet in the air after 3 seconds of elapsed time. This is equivalent to the ordered pair (3, 7) but with the added benefit of knowing which function it is associated with.

### Common Misconceptions or Errors

- Throughout students’ prior experience, two variables written next to one another indicate they are being multiplied. That changes in function notation and will likely cause confusion for some of your students. Continue to discuss the meaning of function notation with these students until they become comfortable with the understanding. In other words, $f$($x$) does not mean $f$ · $x$.
- Students may need additional support in the order of operations.
- For example, for exponential functions, many students multiply $a$ by the growth factor and then raise the product to the value of the exponent.
- For example, students may think that the multiplication is always performed before division.

### Strategies to Support Tiered Instruction

- Instruction is provided to determine the order of operations required once a given input is placed into the function for evaluation. Students may need additional support determining the correct order of operations to perform.
- Teacher models using parentheses to help organize order of operations when evaluating
functions.
- When evaluating $f$($x$) = 4$x$
^{2}for $x$ = −1, teacher can model the use of parentheses by writing the expression 4(−1)^{2}rather than without using parentheses writing 4**⋅**−1^{2 }. This will help students visualize the operations.

- When evaluating $f$($x$) = 4$x$
- Teacher provides instruction for identifying the operations in various functions as they relate to the order of operations using a graphic organizer.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1)*

- The original value of a painting is $9,000 and the value increases by 7% each year. The value
of the painting can be described by the function $V$($t$) = 9000(1 + 0.07)
^{$t$}, where $t$ is the time in years since 1984 and $V$($t$) is the value of the painting.- Part A. Create a table of values that corresponds to this function.
- Part B. Graph the function.

### Instructional Items

*Instructional Item 1*

- Evaluate
*$f$(24), when $f$($x$) = $\frac{\text{3}}{\text{2}}$$x$ + 9.*

Instructional Item 2

Instructional Item 2

- Given $f$($x$) = 2$x$
^{4}− 0.24 $x$^{2 }+ 6.17$x$ − 7, find $f$(2).

**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

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

## STEM Lessons - Model Eliciting Activity

Students are asked to determine a procedure for ranking healthcare plans based on their assumptions and the cost of each plan given as a function. Then, they are asked to revise their ranking based on a new set of data.

The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

## MFAS Formative Assessments

Students are asked to interpret statements that use function notation in the context of a problem.

Students are asked to evaluate a function at a given value of the independent variable.

Students are asked to determine the value of a function, at an input given using function notation, by inspecting its graph.

Students are asked to find the first five terms of a sequence defined recursively, explain why the sequence is a function, and describe its domain

Students are asked to use function notation to rewrite the formula for the volume of a cube and to explain the meaning of the notation.

Students are asked to determine the corresponding input value for a given output using a table of values representing a function, *f*.

Students are asked to determine if each of two sequences is a function and to describe its domain, if it is a function.

## Original Student Tutorials Mathematics - Grades 9-12

Learn how to evaluate and interpret function notation by following Melissa and Jose on their travels in this interactive tutorial.

## Student Resources

## Original Student Tutorial

Learn how to evaluate and interpret function notation by following Melissa and Jose on their travels in this interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Tasks

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

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

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

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

Type: Problem-Solving Task