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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.

Standard #: MA.912.AR.3.4
Standard Information
Standard 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
begin mathsize 12px style f open parentheses x close parentheses end style  2  -1  -2  -1  2
Standard 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
Date of Last Rating: 00/00
Status: State Board Approved
Standard 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 = ax2  + bx + 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 − r1)(x − r2), where  r1 and  r2 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 the connection to completing the square and literal equations to rewrite an equation from standard or factored form to vertex form. 
  • Instruction includes comparing and contrasting between a linear function of the form y = a(x − h) + k and a quadratic function of the form y = a(x − h)2 + k. This will also extend to an absolute value function of the form  y = a |x - h| + k. (MTR.5.1)
  • 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 12 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(12) 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 = 1⁄2 , points 1 unit to the left or right of the vertex are  1⁄2 (12)  or 1⁄2 a unit above or below the vertex. Points 2 units to the left or right of the vertex are 1⁄2  (2)2 or 2 units above or below the vertex. 
    • 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. 
  • 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).

 

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).

 

Strategies to Support Tiered Instruction

  • Teacher emphasizes that because h is subtracted from x in vertex form, it will change the sign of that coordinate. 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. Teachers could show students a graph (technology) of both functions to confirm and make the connection to transformation of functions (MA.912.F.2.1).
  • Teacher models substituting roots into the factored form of a quadratic, y = a(x − r1)(x − r2). Instruction then includes reminding students of different methods that can be used to multiply binomials to convert the quadratic into standard form, y = ax2  + bx + 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) 

  • Part A. Create a quadratic function that contains the roots 3/5 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.
    Table

Instructional Item 3

  • Jayden launches a rocket from the ground that travels in a parabolic path until it lands. After two seconds it reaches a maximum height of 100 feet. The rocket is in the air for five seconds before it hits the ground. Write the quadratic function h(t), representing the height of the rocket above the ground after t seconds.  
 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses
Related Access Points
  • MA.912.AR.3.AP.4 Select a quadratic function to represent the relationship between two quantities from a graph.
Related Resources
Formative Assessment
  • Hotel Swimming Pool Students are asked to write an equation in two variables given a verbal description of the relationship among the variables.
Lesson Plans
  • Solving Quadratic Equations by Completing the square Students will model the process of completing the square (leading coefficient of 1) with algebra tiles, and then practice solving equations using the completing the square method. This lesson provides a discovery opportunity to conceptually see why the process of squaring half of the b value is considered completing the square.
  • 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.
  • 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.
  • Solving Quadratic Equations: Cutting Corners Assess how well students can apply and solve quadratics in one variable. In particular, identify and help students who have difficulties with making sense of a real-life situation and deciding on the math to apply to the problem. Students will be solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and/or factoring. They will also analyze sample responses and interpret results in the context of the situation.
  • Transforming Quadratics Assess how well students can understand different algebraic forms of a quadratic function to reveal properties of its graphical representation. The lesson helps teachers identify and correct misconceptions related to the following: understanding how the factored form of the function can identify a graph's roots, how the completed square form of the function can identify a graph's maximum or minimum point, and how the standard form of the function can identify a graph's intercept.
Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
  • What functions do two graph points determine? This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
  • Graphs of Quadratic Functions Students are challenged to compare and analyze graphs of different quadratic functions, and then use given conditions to produce equations of their own. Students should be familiar with the vertex and intercepts of a parabola as well as the vertex form of the quadratic equation.
Tutorial
  • Quadratic Functions 2 This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.
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.
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