MA.912.AR.5.6

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.1.1 
• MA.912.F.1.6
• MA.912.F.1.8

Terms from the K-12 Glossary

• Coordinate plane
• Domain 
• Exponential Function 
• Function Notation 
• Range 
• $x$-intercept 
• $y$-intercept  

Vertical Alignment

Previous Benchmarks

• MA.8.AR.3

Next Benchmarks

• MA.912.AR.8.2

Purpose and Instructional Strategies

• Instruction provides the opportunity for students to explore the meaning of an asymptote graphically and algebraically. Through work in this benchmark, students will discover asymptotes are useful guides to complete the graph of a function, especially when drawing them by hand. For mastery of this benchmark, asymptotes can be drawn on the graph as a dotted line or not drawn on the graph. 
• For students to have full understanding of exponential functions, instruction includes MA.912.AR.5.3 and MA.912.AR.5.4. Growth or decay of a function can be defined as a key feature (constant percent rate of change) of an exponential function and useful in understanding the relationships between two. 
• Instruction includes the use of $x$-$y$ notation and function notation. 
• Instruction includes representing domain and range using words, inequality notation and set-builder notation. 
  • Words If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number.” 
  • Inequality notation If the domain is all values of $x$ greater than 2, it can be represented as $x$ > 2. 
  • Set-builder notation If the domain is all values of $x$ less than or equal to zero, it can be represented as {$x$|$x$ ≤ 0} and is read as “all values of $x$ such that $x$ is less than or equal to zero.”
• Instruction includes the use of appropriately scaled coordinate planes, including the use of breaks in the $x$- or $y$-axis when necessary.

Common Misconceptions or Errors

• Students may not fully understand how to use proper notation when determining the key features of an exponential function.

Strategies to Support Tiered Instruction

• Instruction includes student understanding that growth and decay is not the same as a function increasing or decreasing. 
  • For example, the exponential function $y$ = −2(0.5)^$x$ is an exponential decay function because the value of $b$ is in between 0 and 1. Note that it is the magnitudes of the $y$-values that are decaying exponentially, eventually getting closer to zero. However, the value of the function increases as the value of $x$ increases. To help students visualize this, graph the function using graphing technology. 
• Instruction includes using an exponential function formula guide like the one below.
  
  Exponential Growth Exponential Decay
  $b$ > 1                 $b$ < 1
  $y$ = $a$(1 + $r$)$t$ $y$ = $a$(1 − $r$)$t$
  

Instructional Tasks

• The bracket system for the NCAA Basketball Tournament (also known as March Madness) is an example of an exponential function. At each round of the tournament, teams play against one another with only the winning teams progressing to the next round. If we start with 64 teams going into round 1, the table of values looks something like this:
  
  
  

• Part A. Graph this function. 
• Part B. What is the percentage of teams left after each round? 

• Ashanti purchased a car for $22,900. The car depreciated at an annual rate of 16%. After 5 years, Ashanti wants to sell her car. 
  • Part A. Write an equation that models the value of Ashanti’s car? 
  • Part B. What would be the range of the graph of the value of Ashanti’s car? 
  • Part C. What would be the $y$-intercept of that graph and what does it represent? 
  • Part D. Will her car ever have a value of $0.00 based on your equation? 
  • Part E. What would be a sensible domain for this function? Justify your answer.

Instructional Items

• An exponential function is given by the equation $y$ = −14($\frac{\text{1}}{\text{4}}$)^$x$. What is the asymptote for 4 the graph? 

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