# MA.912.AR.5.9

Solve and graph mathematical and real-world problems that are modeled with logarithmic functions. Interpret key features and determine constraints in terms of the context.

### Clarifications

Clarification 1: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; end behavior; and asymptotes.

Clarification 2: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Coordinate plane
• Domain
• Exponential function
• Function
• Function notation
• Logarithmic function
• Range
• $x$-intercept
• $y$-intercept

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In Algebra I, students solved problems modeled with linear, exponential and quadratic functions. In Math for College Algebra, students solve problems modeled with logarithmic functions.
• Instruction features a variety of real-world contexts. Some of these contexts should require students to create a function as a tool to determine requested information or should provide the graph or function that models the context.
• Instruction provides the opportunity for students to explore the meaning of an asymptote graphically and algebraically. Through work in this benchmark, students will deepen their understanding of why asymptotes are useful guides to complete the graph of a function. For mastery of this benchmark, asymptotes can be drawn on the graph as a dotted line or not drawn on the graph.
• Instruction includes the use of $x$-$y$ notation and function notation.
• Instruction includes representing domain, range and intervals where the function is increasing, decreasing, positive or negative, using words, inequality notation, set-builder notation and interval 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 range is all values of $y$ less than or equal to zero, it can be represented as {$y$|$y$ ≤ 0} and is read as “all values of $y$ such that $y$ is less than or equal to zero.”
• Interval notation
• If the domain is all values of $x$ less than or equal to 3, it can be represented as (−∞, 3]. If the domain is all values of $x$ greater than 3, it can be represented as (3, ∞). If the range is all values greater than or equal to −1 but less than 5, it can be represented as [−1, 5).
• Depending on a student’s pathway, they may not have worked with interval notation (as that was not an expectation in Algebra I) before this course. Instruction includes making connections between inequality notation and interval notation.
• For example, if the range of a function is −10 < $y$ < 24, it can be represented in interval notation as (−10, 24). This is commonly referred to as an open interval because the interval does not contain the end values.
• For example, if the domain of a function is 0 ≤ $x$ ≤ 11.5, it can be represented in interval notation as [0, 11.5]. This is commonly referred to as a closed interval because the interval contains both end values.
• For example, if the domain of a function is 0 ≤ $x$ < 50, it can be represented in interval notation as [0, 50). This is commonly referred to as a half-open, or half-closed, interval because the interval contains only one of the end values.
• For example, if the range of a function is all real numbers, is can be represented in interval notation as (−∞, ∞). This is commonly referred to as an infinite interval because at least one of end values is infinity (positive or negative).
• Students should be given the opportunity to discuss how constraints can be written and adjusted based on the context they are given.
• Instruction includes differentiating when to use base 10 or natural logs based on the context given.

### Common Misconceptions or Errors

• Students may not fully understand how to interpret proper function notation when determining the key features of a logarithm function.
• Students may not fully understand how to use interval notation.
• When describing intervals where functions are increasing, decreasing, positive or negative, students may represent their interval using the incorrect variable. In these cases, ask reflective questions to help students examine the meaning of the domain and range in the problem.
• Students may miss the need for compound inequalities in their intervals. In these cases, refer to the graph of the function to help them discover areas in their interval that would not make sense in context.

Instructional Task 1 (MTR.2.1, MTR.3.1 and MTR.5.1)
• A video uploaded to social media initially had 75 views one minute after it was posted. The total number of views to date has been increasing exponentially and can be modeled by the function $y$ = 75$e$0.2$t$, where $y$ represents time measured in days since the video was posted.
• Part A. Convert the exponential to logarithmic form.
• Part B. How many days will it take until 5000 people have viewed this video?
• Part C. State the domain for the problem.
• Part D. Sketch a graph that represents days 5 – 10.

### Instructional Items

Instructional Item 1
• Wanda invests \$10,000 in a company with earnings of 4% per year.
• Part A. Write an equation in logarithmic form that describes the situation.
• Part B. How long will it take to accumulate \$20,000?
• Part C. How long will it take to accumulate \$25,000?
• Part D. Sketch a graph that represents the amount accumulated from the initial investment to reaching the goal of \$25,000.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1202340: Precalculus Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))
1209315: Mathematics for ACT and SAT (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.AR.5.AP.9: Given a mathematical and/or real-world problem that is modeled with logarithmic functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Type: Perspectives Video: Professional/Enthusiast

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert