MA.912.AR.5.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

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

Terms from the K-12 Glossary

• Exponential Function

Vertical Alignment

Previous Benchmarks

• MA.7.AR.3

Next Benchmarks

• MA.912.AR.5.5

Purpose and Instructional Strategies

• Provide opportunities to reference MA.912.AR.1.1 as students identify and interpret parts of an exponential equation or expression as growth or decay. 
• Instruction includes the connection to growth or decay of a function as a key feature (constant percent rate of change) of an exponential function and being useful in understanding the relationships between two quantities. 
• Instruction includes the use of graphing technology to explore exponential functions. 
  • For example, students can explore the function $f$($x$) =  $a$$b$^$x$ and how the $a$-value and $b$-value are affected. Ask questions like “What impact does changing the value of $a$ have on the graph? What about $b$? What values for $b$ cause the function to increase? Which values cause it to decrease?” As students explore, formalize the terms exponential growth and decay when appropriate. 
  • As students explore the graph, have students choose values of $a$ and $b$ to complete a table of values. Once completed ask students what causes the value of $y$ to increase or decrease as the value of $x$ increases. Guide students to see that it’s because $b$ > 1 or $b$ < 1. Have students adjust the graph and repeat this exercise. 
  • Once students have an understanding of what causes exponential growth and decay, both graphically and algebraically, ask students if they think the curve ever passes $y$ = 0. Have students extend their function table for exponential decay to include more extreme values for $x$ to explore if it ever does. As student arrive at an understanding that it does not cross $y$ = 0, guide them to understand why it doesn’t algebraically. Once students arrive at this understanding, define this boundary as an asymptote. 
  • As students explore the provided graph, they will move the slider for $b$ to have negative values. The resulting graphs provide an interesting discussion point. Have students complete a function table using negative $b$ values. Students should quickly see the connection between the two “curves” and why neither is continuous. Let students know that for this reason, most contexts for exponential functions restrict $b$ to be greater than 0 and not equal to 1. 
• As students solidify their understanding of $f$($x$) = $a$$b$^$x$, use graphing technology again to have them explore the form $f$($x$) = $x$(1 ± $r$)^$r$  . Guide students to use the sliders for $a$ and $r$ to visualize that only $r$ determines whether the function represents exponential growth or exponential decay. 
  • Have students discuss which values for $r$ cause exponential growth or decay. They should observe that negative values cause exponential decay while positive values cause exponential growth.

Common Misconceptions or Errors

• Students may not understand exponential function values will eventually get larger than those of any other polynomial functions because they do not fully understand the impact of exponents on a value. 
• Students may not understand that growth factors have one constraint ($b$ > 1) while decay factors have a compound constraint (0 < $b$ < 1). Some students may think that as long as $b$ < 1, the function will represent exponential decay. 
• Students may think that if $a$ is negative and $r$ > 0 or $b$ > 1, the function represents an exponential decay. To address this misconception, help students understand that the negative values are growing at an exponential rate.

Strategies to Support Tiered Instruction

• Teacher provides instruction to identify exponential functions in all methods (i.e., graphs, equations and tables). 
  • For example, instruction may include providing a comparison of the two forms of exponential functions. Having a side-by-side comparison of both as an equation, graph and a table of values will provide a visual aid. 
• Teacher provides student with examples and non-examples of exponential functions in tables. 
• For example, teacher can provide the following tables for students to determine which ones represent an exponential function.
  
  
  

Instructional Tasks

• After a person takes medicine, the amount of drug left in the person’s body changes over time. When testing a new drug, a pharmaceutical company develops a mathematical model to quantify this relationship. To find such a model, suppose a dose of 1000 mg of a certain drug is absorbed by a person’s bloodstream. Blood samples are taken every five hours, and the amount of drug remaining in the body is calculated. The data collected from a particular sample is recorded below. 

• Part A. Does this data represent an exponential growth or decay function? Justify your answer. 
• Part B. Create an exponential function that describes the data in the table above.

Instructional Items

• Given the function $f$($x$) = 125(1 − 0.26)^x , does it represent an exponential growth or decay function?

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