Given the graph or table of f(x) and the graph or table of f(x)+k,kf(x), f(kx) and f(x+k), state the type of transformation and find the value of the real number k.

### Clarifications

*Clarification 1:*Within the Algebra 1 course, functions are limited to linear, quadratic and absolute value.

General Information

**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

- MA.912.AR.2.4
- MA.912.AR.2.5
- MA.912.AR.3.7
- MA.912.AR.3.8
- MA.912.AR.4.4
- MA.912.AR.5.6
- MA.912.AR.5.7
- MA.912.AR.5.8
- MA.912.AR.5.9
- MA.912.F.1.1

### Terms from the K-12 Glossary

- Transformation
- Translation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students performed single transformations on two-dimensional figures. In Algebra I, students identified the effects of single transformations on linear, quadratic and absolute value functions. In Geometry, students performed multiple transformations on two-dimensional figures. In Mathematics for College Algebra, students determine the type of transformations on linear, quadratic, exponential, logarithmic and absolute value functions.- Instruction includes identifying function transformations involving a combination of translations, dilations and reflections, and determining the value of the real number that defines each of the transformations.
- Transformations can be either horizontal (changes to the input: $x$) or vertical (changes to the output: $f$($x$)).
- There are three different types of transformations: translations, dilations, and reflections.
- By combining single transformations, a parent function can become a more advanced function. $f$($x$) = $x$ → $h$($x$) = $A$$f$($B$($x$ + $C$) + $D$)Example: $f$($x$) = $x$
^{2}^{2}+ 5

- By combining single transformations, a parent function can become a more advanced function.

- Using a graphing utility can help students understand how changing the value of the real numbers in the function’s equation change its graph.
- Encourage student’s discussion about the effects of changing the value of the real numbers $A$, $B$, $C$ and $D$ in the equation of the function. Ask them to generalize their findings.

### Common Misconceptions or Errors

- Some students may have difficulty seeing the impact of a transformation when comparing tables and graphs. In these cases, encourage students to convert the graph to a second table, using the same domain as the first table. This should aid in comparisons.
- Similar to writing functions in vertex form, students may confuse effect of the sign of $k$ in $f$($x$ + $k$). Direct these students to examine a graph of the two functions to see that the horizontal shift is opposite of the sign of $k$.
- Vertical stretch/compression can be hard for students to see on linear functions initially and they may interpret stretch/compression as rotation. Introduce the effects of $k$$f$($x$) and $f$($k$$x$) by using a quadratic or absolute value function first before analyzing the effect on a linear function.
- Students may think that a vertical and horizontal stretch from $k$$f$($x$) and $f$($k$$x$) look the same. For linear and quadratic functions, it can help to have a non-zero $y$-intercept to visualize the difference.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1)*

- A graph and table, which represents an absolute value function, are shown below. Describe and determine the value of the real number that defines the transformation from $f$($x$) to $g$($x$).

*Instructional Task 2 (*

*MTR.3.1*)- Describe the transformations that maps the function $f$($x$) = 2$x$ to each of the following. $f$($x$) = 2$x$-2 $g$($x$) = 4$x$+ 3
*$h$($x$) =*2$x$ − 3*$b$($x$)= 3(*2$x$ + 1*)+2*

### Instructional Items

*Instructional Item 1*

- Considering the graph of $f$($x$) and $g$($x$) below, describe the transformation and determine the value of the real number, $k$, that defines the transformation from $f$($x$) to $g$($x$).

*Instructional Item 2 *

- Considering the table below, describe the transformation and determine the value of the real number, $k$, that defines the transformation from $f$($x$) to $g$($x$).

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

## Related Courses

This benchmark is part of these courses.

1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))

1209315: Mathematics for ACT and SAT (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

MA.912.F.2.AP.3:

Given the graph of a given function after replacing f(x) by f(x) + k and f(x + k), kf(c), for specific values of k select the type of transformation and find the value of the real number k.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessment

## Lesson Plans

## Original Student Tutorial

## MFAS Formative Assessments

Write the Equations:

Students are given the graphs of three absolute values functions and are asked to write the equation of each.

## Original Student Tutorials Mathematics - Grades 9-12

Dilations...The Effect of k on a Graph:

Visualize the effect of using a value of k in both *kf*(*x*) or *f*(*kx*) when k is greater than zero in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Dilations...The Effect of k on a Graph:

Visualize the effect of using a value of k in both *kf*(*x*) or *f*(*kx*) when k is greater than zero in this interactive tutorial.

Type: Original Student Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.