MA.912.F.2.3

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

• 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 → $g$($x$) = 2($x$ + 3)² + 5

• 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

• 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 $g$($x$). 

• 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 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 $g$($x$).

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