MA.912.F.2.4

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. 

• Using a graphing utility can help students understand how changing the value of the real numbers in the function’s equation change its graph (MTR.5.1). 
• 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 (MTR.4.1). Ask them to generalize their findings. 

• Instruction includes determining the order for performing two or more transformations to a single function. A different order of transformations may result in a different outcome. 
  • It is helpful to relate the order of the transformations to the order of operations. 
    • Horizontal shifts 
    • Vertical stretching or shrinking 
    • Reflecting 
    • Vertical shift

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). 
• Similar to writing functions in vertex form, students may confuse the effect of the sign of $k$ in ($x$ + $k$). 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 $k$($x$) and $f$($k$$x$) by using a quadratic or absolute value function first before analyzing the effect on a linear function. 
• Students may think that a vertical and horizontal stretch from $k$($x$) and $f$($k$$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

• Below are the graphs of $f$($x$) and $g$($x$). 

• Part A. Describe the sequence of transformations that map the graph of $f$($x$) to the graph of $g$($x$). 
• Part B. If $f$($x$) = $\sqrt{\mathrm{x}}$, write an equation for $g$($x$). 

• Part A. Describe the transformation of $f$($x$) represented in Step 1. State the value of $B$. 
• Part B. Describe the transformation of $f$($x$) represented in Step 2. State the value of $A$. 
• Part C. Describe the transformation of $f$($x$) represented in Step 3. State the value of $D$. 
• Part D. Write an equation for $g$($x$) in terms of $f$($x$).

Instructional Item 2 

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