# MA.912.AR.3.8

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

### Examples

Algebra 1 Example: The value of a classic car produced in 1972 can be modeled by the function , where t is the number of years since 1972. In what year does the car’s value start to increase?

### Clarifications

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

Clarification 2: Instruction includes the use of standard form, factored form and vertex form.

Clarification 3: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation.

Clarification 4: Within the Algebra 1 course, notations for domain, range and constraints are limited to inequality and set-builder.

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
• Function Notation
• Range
• $x$-intercept
• $y$-intercept

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, solved problems involving real-world linear equations. In Algebra I, solve problems that are modeled by quadratic functions. Students additionally graph the function and determine or interpret key features of the function. In later courses, this work expands to exponential and other kinds of functions.
• This benchmark is a culmination of MA.912.AR.3. Instruction here should feature 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 includes making connections to various forms of quadratic equations to show their equivalency. Students should understand and interpret when one form might be more useful than other depending on the context.
• Standard Form
Can be described by the equation $y$ = $a$$x$2 + $b$$x$ + $c$, where $a$, $b$ and $c$ are any rational number. This form can be useful when identifying the y-intercepts.
• Factored Form
Can be described by the equation $y$ = $a$($x$$r$1) ($x$$r$2), where $r$1 and $r$2 are real numbers and the roots, or $x$-intercepts. This form can be useful when identifying the $x$-intercepts, or roots.
• Vertex Form
Can be described by the equation $y$ = $a$($x$$h$)2+ $k$, where the point ($h$, $k$) is the vertex. This form can be useful when identifying the vertex.
• Instruction includes the use of $x$-$y$ notation and function notation.
• Instruction includes representing domain, range and constraints 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 provides opportunities to make connections between the domain and range and other key features.
• For example, a coffee shop uses the function, $P$($x$) =  −80$x$+ 480$x$ − 540 to model the profit they can earn in thousands of dollars in terms of the price per cup of coffee, in dollars. By determining the domain and range that includes prices that yield a positive profit, one would also have to identify the vertex (or maximum) and the roots of the function. Students should realize that they can do this by transforming the given expression into vertex form.

### Common Misconceptions or Errors

• Students may find themselves stuck initially, unsure of where to start. In conversations with these students, prompt them to reflect on what they know about the context and how they can use that information to determine the requested information (MTR.1.1).
• For example, students may have an equation in standard form and need to interpret the vertex in context. Prompting students to consider how they’ve calculated vertices in the past should lead them to choose to either convert the equation into vertex form or use the line of symmetry to help determine the vertex.

### Strategies to Support Tiered Instruction

• Teacher provides equations in both function notation and $x$-$y$ notation and models graphing both forms using a graphing tool or graphing software (MTR.2.1).
• For example, $f$($x$) = ($x$ − 2.3)2 + 7 and =($x$ − 2.3)2 + 7, to show that both $f$($x$) and $y$ represent the same outputs of the function.
• Instruction provides opportunities to visualize the domain and range on a graph using a highlighter.
• For example, a coffee shop uses the function $P$($x$) = −80$x$+ 480$x$ − 540 to model the profit they can earn in thousands of dollars in terms of the price per cup of coffee, in dollars. If the coffee shop is only interested in prices that yield a positive profit, the highlighted domain and range are shown.

•  ABC Pool Company is constructing a 17 feet by 11 feet rectangular pool. Along each side of the pool, they plan to construct a concrete sidewalk that has a constant width. The land parcel being used has a total area of 315 sq. ft. to construct the pool and sidewalks. The function $f$($x$) = 4$x$+ 56$x$ − 128, represents the situation described. Transform the function to determine and interpret its zeros.

### Instructional Items

Instructional Item 1
• A new coffee shop wants to maximize their profit within the first year of business. They determined the function, $P$($x$) = −80$x$2 + 480$x$ − 540, models the profit they can earn in thousands of dollars in terms of the price per cup of coffee, in dollars. What price of coffee maximizes the coffee shop’s profit?

Instructional Item 2

• The value of a classic car produced in 1972 can be modeled by the function $V$($t$) = 19.25$t$2− 440$t$ + 3500, where $t$ is the number of years since 1972.
• Part A. In what year does the car’s value start to increase?
• Part B. Will the car’s will ever begin to decrease? Assume it follows its modeled value.

*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.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
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))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 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.3.AP.8: Given a mathematical and/or real-world problem that is modeled with quadratic 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.

## Formative Assessments

Model Rocket:

Students are asked to graph a function in two variables given in context.

Type: Formative Assessment

Rocket Town:

Students are asked to rewrite a quadratic expression in vertex form to find maximum and minimum values.

Type: Formative Assessment

Jumping Dolphin:

Students are asked to find the zeros of a quadratic function in the context of a modeling problem.

Type: Formative Assessment

## Lesson Plans

Discovering Properties of Parabolas by Comparing and Contrasting Parabolic Equations:

• Teachers can use this resource to teach students how to derive the equation of a parabola in vertex form y = a(x – h)2 + k, when given the (x, y) coordinates of the focus and the linear equation of the directrix.
• An additional interactive graphing spreadsheet can be used as a resource to aid teachers in providing examples.

Type: Lesson Plan

Acting Out A Parabola: the importance of a vertex and directrix:

Students will learn the significance of a parabola's vertex and directrix. They will learn the meaning of what exactly a parabola is by physically representing a parabola, vertex, and directrix. Students will be able to write an equation of a parabola given only a vertex and directrix.

Type: Lesson Plan

How High Can I Go?:

Students will graph quadratic equations, and identify the axis of symmetry, the maximum/minimum point, the vertex, and the roots. Students will work in pairs and will move around the room matching equations with given graphs.

Type: Lesson Plan

Ranking Sports Players (Quadratic Equations Practice):

In this Model Eliciting Activity, MEA, students will rank sports players by designing methods, using different indicators, and working with quadratic equations.

Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.

Type: Lesson Plan

Parts and more Parts-- Parabola Fun:

This is an entry lesson into quadratic functions and their shapes. Students see some real-life representations of parabolas. This lesson provides important vocabulary associated with quadratic functions and their graphs in an interactive manner. Students create a foldable and complete a worksheet using their foldable notes.

Type: Lesson Plan

## Perspectives Video: Experts

Type: Perspectives Video: Expert

Using Mathematics to Optimize Wing Design:

Nick Moore discusses his research behind optimizing wing design using inspiration from animals and how they swim and fly.

Type: 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

<p>Get in gear with robotics as this engineer explains how quadratic equations are used in programming robotic navigation.</p>

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Ideas

Solving Quadratic Equations using Babylonian Multiplication:

Unlock an effective teaching strategy for teaching solving quadratic equations using Babylonian multiplication in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Solving Quadratic Equations by Completing the Square:

Unlock an effective teaching strategy for teaching solving quadratic equations by completing the square in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Making Connections with Vieta's Formula:

Unlock an effective teaching strategy for explaining the equation of the axis of symmetry using Vieta's formula in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Solving Quadratic Equation Using Loh's Method:

Unlock an effective teaching strategy for solving quadratic equations in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

## Tutorial

This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.

Type: Tutorial

## STEM Lessons - Model Eliciting Activity

Ranking Sports Players (Quadratic Equations Practice):

In this Model Eliciting Activity, MEA, students will rank sports players by designing methods, using different indicators, and working with quadratic equations.

Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.

## MFAS Formative Assessments

Jumping Dolphin:

Students are asked to find the zeros of a quadratic function in the context of a modeling problem.

Model Rocket:

Students are asked to graph a function in two variables given in context.

Rocket Town:

Students are asked to rewrite a quadratic expression in vertex form to find maximum and minimum values.

## Student Resources

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

## Perspectives Video: Experts

Type: 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

<p>Get in gear with robotics as this engineer explains how quadratic equations are used in programming robotic navigation.</p>

Type: Perspectives Video: Professional/Enthusiast

## Tutorial

This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.

Type: Tutorial

## 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