### 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.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Algebraic Reasoning

**Standard:**Write, solve and graph quadratic equations, functions and inequalities in one and two variables.

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Coordinate Plane
- Domain
- Function Notation
- Quadratic Function
- 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.

- Standard Form
- 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.”

- Words
- 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$x$
^{2 }+ 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.

- For example, a coffee shop uses the function, $P$($x$) = −80$x$

### 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.

- For example, $f$($x$) = ($x$ − 2.3)
- 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$x$
^{2 }+ 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.

- For example, a coffee shop uses the function $P$($x$) = −80$x$

### Instructional Tasks

*Instructional Task 1 (*

*MTR.2.1*, MTR.5.1)- 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$x$
^{2 }+ 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

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.*

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## Lesson Plans

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## Perspectives Video: Professional/Enthusiast

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## Tutorial

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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

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

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

## Student Resources

## Perspectives Video: Experts

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Type: Perspectives Video: Expert

<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

## Perspectives Video: Expert

<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