Write a quadratic function to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

### Examples

*Algebra I Example:*Given the table of values below from a quadratic function, write an equation of that function.

x | -2 | -1 | 0 | 1 | 2 |

2 | -1 | -2 | -1 | 2 |

### Clarifications

*Clarification 1*: Within the Algebra 1 course, a graph, written description or table of values must include the vertex and two points that are equidistant from the vertex.

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

*Clarification 3*: Within the Algebra 2 course, one of the given points must be the vertex or an *x*-intercept.

General Information

**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 middle grades, students wrote linear two-variable linear equations. In Algebra I, students write quadratic functions from a graph, written description or table. In later courses, students will write quadratic two-variable inequalities.- Instruction includes making connections to various forms of quadratic equations to show
their equivalency. Students should understand 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 $y$-intercept. - 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
Can be described by the equation $y$ = $a$$x$
- Instruction includes the use of $x$-$y$ notation and function notation.
- Instruction includes the connection to completing the square and literal equations to rewrite an equation from standard or factored form to vertex form.
- When determining the value of $a$ in a quadratic function, this can be done by two
methods described below.
- Students may notice a pattern from the points in the graph. When $a$ = 1, points 1
unit to the left or right of the vertex are 1
^{2}or 1 unit above or below the vertex. Points 2 units to the left or right of the vertex are (2)^{2 }or 4 units above or below the vertex. Students may look at the table and notice this relationship exists, therefore, $a$ = 1. - This process can be used for other values of $a$. When $a$ = 2, for example,
points 1 unit to the left or right of the vertex are 2(1
^{2}) or 2 units above or below the vertex. Points 2 units to the left or right of the vertex are 2(2)^{2}or 8 units above or below the vertex. Similarly, when $a$ = $\frac{\text{1}}{\text{2}}$, points 1 unit to the left or right of the vertex are $\frac{\text{1}}{\text{2}}$(1^{2}) or $\frac{\text{1}}{\text{2}}$ a unit above or below the vertex. Points 2 units to the left or right of the vertex are $\frac{\text{1}}{\text{2}}$ (2)^{2}or 2 units 2 above or below the vertex.

- This process can be used for other values of $a$. When $a$ = 2, for example,
points 1 unit to the left or right of the vertex are 2(1
- Students can solve for a by substituting the values of $x$ and $y$ from a point on the
parabola.
- Ask students to consider $y$ = $a$($x$ + 4)
^{2}− 2 and determine what information they would need to be able to solve for a. As students express the need to know values for $x$ and for $y$, ask them if they know any combinations of $x$ and $y$ that are solutions. Students could use a table of values or a graph, if given, to determine values that could be used for $x$ and $y$. Have students pick one and substitute and solve for $a$. Ask for students who chose different points to share their value for a to help them see that all points, when substituted, produce $a$ = 1.

- Ask students to consider $y$ = $a$($x$ + 4)

- Students may notice a pattern from the points in the graph. When $a$ = 1, points 1
unit to the left or right of the vertex are 1
- Instruction includes the use of graphing software or technology.
- For example, when determining the value of $a$, consider using graphing software
to allow students to use sliders to quickly observe this. This provides opportunity
for students to notice patterns regarding the value of $a$ and the concavity and
stretch of the parabola
*(MTR.5.1).*

- For example, when determining the value of $a$, consider using graphing software
to allow students to use sliders to quickly observe this. This provides opportunity
for students to notice patterns regarding the value of $a$ and the concavity and
stretch of the parabola

### Common Misconceptions or Errors

- When writing functions in vertex form, students may confuse the sign of $h$.
- For example, students may see a vertex of (−1, −2) and an $a$ value of 3 and write
the function as $y$ = 3($x$ − 1)
^{2 }− 2 instead of $y$ = 3($x$ + 1)^{2}− 2. To address this, help students recognize that because $h$ is subtracted from $x$ in vertex form, it will change the sign of that coordinate. Show students a graph of both functions to confirm and make the connection to transformation of functions (MA.912.F.2.1).

- For example, students may see a vertex of (−1, −2) and an $a$ value of 3 and write
the function as $y$ = 3($x$ − 1)

### Strategies to Support Tiered Instruction

- Teacher models substituting roots into the factored form of a quadratic, $y$ = $a$($x$ − $r$
_{1})($x$ − $r$_{2}). Instruction then includes reminding students of different methods that can be used to multiply binomials to convert the quadratic into standard form, $y$ = $a$$x$^{2 }+ $b$$x$ + $c$. - Instruction includes solving for a by substituting the values of $x$ and $y$ from a point on
the parabola. Students may need to review the order of operations to ensure they correctly
isolate $a$. Provide students with a review of the order of operations.
- Parenthesis
- Exponents
- Multiplication or Division (Whatever comes first left to right)
- Addition or Subtraction (Whatever comes first left to right)

- Instruction includes recall of knowledge demonstrating how when $h$ is subtracted from $x$ in vertex form, it will change the sign of that coordinate. A graph of both functions can be used to confirm and make the connection to transformation of functions.
- Teacher provides a laminated cue card of the steps required to convert from factored form to standard form.

### Instructional Tasks

*Instructional Task 1 (MTR.7.1)*

- Duane throws a tennis ball in the air. After 1 second, the height of the ball is 53 ft. After 2
seconds, the ball reaches a maximum height of 69 feet. After 3 seconds the height of the ball
is 53 ft.
- Part A. Write a quadratic function to represent the height of the ball, $h$, at any point in time, $t$.
- Part B. How long will the tennis ball stay in the air? Round your answer to the nearest tenth second.

Instructional Task 2 (MTR.4.1, MTR.5.1)

Instructional Task 2 (MTR.4.1, MTR.5.1)

- Part A. Create a quadratic function that contains the roots $\frac{\text{3}}{\text{5}}$ and 1.8.
- Part B. Compare your function with a partner. What do you notice?

### Instructional Items

*Instructional Item 1*

- Write the quadratic function that corresponds with the graph below.

*Instructional Item 2*

- Given the table of values below from a quadratic function, write an equation of that function.

**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))

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))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

MA.912.AR.3.AP.4: Select a quadratic function to represent the relationship between two quantities from a graph.

## Related Resources

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

## Formative Assessment

## Perspectives Video: Professional/Enthusiast

## MFAS Formative Assessments

Hotel Swimming Pool:

Students are asked to write an equation in two variables given a verbal description of the relationship among the variables.

## Student Resources

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

## Parent Resources

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