MA.912.F.2.5

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

• MA.8.GR.2 
• MA.912.GR.2

 

Next Benchmarks

• MA.912.F.1.7

 

Purpose and Instructional Strategies

• 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$) is a horizontal stretch or compression of $f$($x$). Multiplying all the inputs ($x$-values) of a function by a real number, $B$, will result in a horizontal stretching or compression depending on the value of $B$. If $B$ is between 0 and 1 (0 < $B$ <1), the graph will be horizontally stretched by $\frac{\text{1}}{\text{<math><mi>B</mi></math>}}$ and if $B$ is greater than 1 ($B$ > 1), the graph will be horizontally compressed by $\frac{\text{1}}{\text{<math><mi>B</mi></math>}}$ 
• 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

• 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

• 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 $g$($x$)? 

• 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 $f$($x$). 

• A function $f$($x$) is given. Create a table for the functions below 
  a. $g$($x$) = $f$($x$) + 5
  b. $h$($x$) = $f$(2$x$)
  
  
  

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