Given a table, equation or graph that represents a function, create a corresponding table, equation or graph of the transformed function defined by adding a real number to the

*x*- or*y*-values or multiplying the*x*- or*y*-values by a real number.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 identify effects of transformations on linear, quadratic, exponential, logarithmic and absolute value functions.- In this benchmark, students will create a table, equation or graph of a transformed function defined by adding a real number to the $x$-or $y$-values or multiplying the $x$-or $y$-values by a real number.
- Instruction includes the use of a graphic software to ensure adequate time for students to examine multiple transformations on the graphs of functions.
- Given a function $f$, the transformed function $g$($x$) = $f$($x$ − $C$) is a horizontal shift of $f$($x$). Adding a real number, $C$$x$, to all the inputs ($x$-values) of a function will result in shifting the output left or right depending on the sign of $C$. If $C$> is positive, the graph will shift right. If $C$ is negative, the graph will shift left.

- Given a function $f$, the transformed function $g$($x$) = $f$($x$) + $D$ is a vertical shift of $f$($x$). Adding a real number, $D$, to all the outputs ($y$-values) of a function will result in shifting the output up or down depending on the sign of $D$. If $D$ is positive the graph will shift up, and if $D$ is negative the graph will shift down.

- Discuss with the students that as well as translations of two-dimensional figures, adding a constant to either the input or output of a function change the position of the graph, but it doesn’t change the shape of the graph
*(MTR.4.1)*. - Given a function $f$, the transformed function $g$($x$) = $A$$f$($x$) is a vertical stretch or compression of $f$($x$). Multiplying all the outputs ($y$-values) of a function by a real number, $A$, will result in a vertical stretching or compression depending on the value of $A$. If $A$ is between 0 and 1 (0 < $A$ < 1), the graph will be vertically compressed and if $A$ is greater than 1 ($A$ > 1), the graph will be vertically stretched.
- If $A$ is a negative number ($A$ < 0), the transformed graph will be a combination of a vertical stretch or compression and a reflection over the $x$-axis. Discuss with students how multiplying all the $y$-values by −1 is the same as reflecting a two-dimensional figure over the $x$-axis
*(MTR.4.1)*.

- Given a function $f$, the transformed function
*$g$($x$) =*$B$$f$($x$) - If $B$ is a negative number ($B$ < 0), the transformed graph will be a combination of a horizontal stretch or compression and a reflection over the $y$-axis. Discuss with students how multiplying all the $x$-values by −1 is the same as reflecting a two-dimensional figure over the $y$-axis
*(MTR.4.1)*.

- Discuss with students the meaning of $g$($x$) = $f$(2$x$). In this case, the output value, $g$($x$), is the same as the output value of $f$($x$) at an input that is twice the size.

- Discuss with students the meaning of $g$($x$) = $f$(($\frac{\text{1}}{\text{2}}$)$x$). In this case the output value, $g$($x$) is the same as the output value of $f$($x$) at an input that is half the size. Example: $g$(4) = $f$($\frac{\text{1}}{\text{2}}$ · 4) = $f$(2)=4
*(MTR.4.1)*.

### 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
*(MTR.2.1)*. - Some students misinterpret how the parameters of the equation of a transformed function are affected by a horizontal translation. This may indicate that students do not understand the relationship between the graph and the equation of the function.
- For example, a student may think that $g$($x$) = $f$($x$ + 1) is a horizontal translation to the right because of the positive addend for $x$. One potential teaching strategy would be using a graphing utility to graph the function $f$($x$) = ($x$ − C)2 creating $C$ as slider, and then allowing students to explore the translation results as the value of the slider changes.

- Some students may have difficulties understanding that multiplying the input of a function by a number greater than 1 will result in a horizontal compression of the graph instead of a stretching. It is important to point out that multiplying the $x$-value does not change the original value of the input. Because the input is being multiplied by a number greater than 1, a smaller input in the transformed function is needed to obtain the same output from the original function. One potential teaching strategy would be using a graphing utility to graph the function $f$($x$) = ($B$$x$)2 creating $B$ as slider, and then allowing students to explore the stretching/compression results as the value of the slider changes from 0 to 2. Remind students that negative values of $B$ will result in a vertical reflection of the function.

### Instructional Tasks

*Instructional Task 1 (*

*MTR.2.1*)- The figure shows the graph of a function $f$ whose domain is the interval −4 ≤ $x$ ≤4.

- Part A: Sketch the graph of each transformation described below and compare it with the graph of $f$. Explain what you see.

a. $g$($x$) = $f$($x$) + 2

b. $h$($x$) = $f$($x$ + 2)

c. $k$($x$) = 2$f$($x$)

d. $r$($x$) = $f$(2$x$) - Part B: The points labeled $M$, $N$, $P$ on the graph of $f$ have coordinates $M$ = (−4, −5), $N$ = (0,−1,) and $P$ = (−4,4). Complete the table below with the coordinates of the points corresponding to $M$, $N$, $P$ on the graphs of $g$, $h$, $k$ and $r$?

### Instructional Items

*Instructional Item 1 (MTR.3.1)*

- Given the function $f$($x$) = |$x$|, graph the function $f$($x$) and the transformation $g$($x$) = $f$($x$ − 3) on the same axes. What do you notice about the $x$-intercepts of $g$($x$)?

Instructional Item 2 (

Instructional Item 2 (

*MTR.3.1*)- Given the function $f$($x$) = log x, graph the function $f$($x$) and the transformation $g$($x$) = 3$f$($x$) on the same axes. Describe the transformed function, $g$($x$), as it relates to the graph of $f$($x$).

Instructional Item 3 (

Instructional Item 3 (

*MTR.3.1*)- A function $f$($x$) is given. Create a table for the functions below

a.*$g$($x$) = $f$($x$)*+ 5

b. $h$($x$) = $f$(2$x$)

**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.

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))

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))

## Related Access Points

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

MA.912.F.2.AP.5: Given a table, equation or graph that represents a function, select a corresponding table, equation or graph of the transformed function defined by adding a real number to the x- or y-values.

## Related Resources

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

## Original Student Tutorial

## 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.