algebra 1 unit 1 lesson 6 homework

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Get ready for Algebra 1

Unit 1: get ready for equations & inequalities, unit 2: get ready for working with units, unit 3: get ready for linear relationships, unit 4: get ready for functions & sequences, unit 5: get ready for exponents, radicals, & irrational numbers, unit 6: get ready for quadratics.

Common Core Algebra I Math (Worksheets, Homework, Lesson Plans)

Related Topics: Common Core Math Resources, Lesson Plans & Worksheets for all grades Common Core Math Video Lessons, Math Worksheets and Games for Algebra Common Core Math Video Lessons, Math Worksheets and Games for all grades

Looking for video lessons that will help you in your Common Core Algebra I math classwork or homework? Looking for Common Core Math Worksheets and Lesson Plans that will help you prepare lessons for Algebra I students?

The following lesson plans and worksheets are from the New York State Education Department Common Core-aligned educational resources. Eureka/EngageNY Math Algebra I Worksheets.

These Lesson Plans and Worksheets are divided into five modules.

Algebra I Homework, Lesson Plans and Worksheets

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Vertex Form

15.1: Notice and Wonder: Two Sets of Equations (5 minutes)

CCSS Standards

Building Towards

• HSF-IF.C.8.a

Routines and Materials

Instructional Routines

• Notice and Wonder

This warm-up prompts students to analyze two sets of equations that they will study more closely in a later activity. In each set, the three equations define the same function but are written in different forms—factored form, standard form, and vertex form. Noticing and wondering about the features of the equations prepares students to reason later that the expressions defining each output are equivalent.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language used to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Display the two sets of equations for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions. Follow with a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

\(f(x)= x^2 +4x\)

\(g(x)=x(x+4)\)

\(h(x)=(x+2)^2 -4\)

\(p(x)=\text-x^2+6x-5\)

\(\\q(x)=(5-x)(x-1)\)

\(r(x)=\text-1(x-3)^2+4\)

Student Response

For access, consult one of our IM Certified Partners .

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the reasoning on or near the relevant equations. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information, etc. 

15.2: A Whole New Form (10 minutes)

This activity introduces students to the vertex form . Students examine the parameters in expressions of this form and the graphs of functions defined by such expressions. They look for structure in the given representations and notice that there are connections between the numbers in each expression and the vertex of the corresponding graph (MP7).

Students also see that we can rewrite the expression in vertex form into another form and show that the expressions are equivalent. Writing equivalent expressions allows students to practice applying the distributive property to expand expressions containing two sums or two differences.

Give students a minute to read the task statement. Before students answer the questions, briefly discuss the worked example to make sure students can follow the reasoning that illustrates the equivalence of the expressions defining \(h\) and \(f\) .

Here are two sets of equations for quadratic functions you saw earlier. In each set, the expressions that define the output are equivalent. 

\(q(x)=(5-x)(x-1)\)

The expression that defines \(h\) is written in vertex form . We can show that it is equivalent to the expression defining \(f\) by expanding the expression:

\(\displaystyle \begin {align} (x+2)^2-4 &=(x+2)(x+2)-4\\ &=x^2+2x+2x+4-4\\ &=x^2+4x\\ \end{align}\)

• Show that the expressions defining \(r\) and \(p\) are equivalent.

Graph of \(h\)

Graph of \(r\)

Anticipated Misconceptions

Some students may incorrectly think that \((x-3)^2\) is \((x^2-3^2)\) . Remind them that \((x-3)^2\) means \((x-3)(x-3)\) . Consider pointing out the example in the task statement to help students show that the expressions for  \(r\) and \(p\) are equivalent.

Invite students to show that the expressions defining functions  \(r\) and \(p\)  are equivalent. Consider pointing out that at the moment it is easier to show equivalence by going from vertex form to standard form than from standard form to vertex form. In a future unit, we will look at how to do the latter.

Then, discuss questions such as:

• “What information does a quadratic expression in the vertex form reveal? How does it show that information?” (It reveals the coordinates of the vertex of the parabola. The number in the parentheses seems to be related to the \(x\) -coordinate of the vertex. The number outside seems to be the \(y\) -coordinate of the vertex.)
• “What doesn’t it tell us?” (It does not allow us to easily see the \(x\) - or \(y\) -intercepts.)
• “Why do you think this form is used?” (Sometimes we want to know the maximum or the minimum of a function. It is helpful to be able to see it in the expression or equation defining the function.)
• “Can you give an example of a situation when it might be useful to have the relationship modeled using an expression in vertex form?” (Examples: when we are interested in the maximum height of an object in projectile motion, or when we want to know the maximum revenue in a business model.)

15.3: Playing with Parameters (20 minutes)

• HSF-IF.C.7.a
• MLR8: Discussion Supports

Required Materials

• Graphing technology

Earlier, students noticed that the numbers in a quadratic expression in vertex form are related to the coordinates of the vertex of the graph. Here, they investigate those connections closely. Just as they have done with expressions in standard and factored forms, students use technology to experiment with each parameter of expressions in vertex form and study the effects on the graph. In this process, they practice looking for regularity in repeated reasoning (MP8). If working with a partner, students will take turns using the graphing technology and recording observations. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

The work also encourages students to begin looking at the structure of the form—a squared expression with a coefficient \(a\) and a constant term (MP7). In an upcoming lesson, they will further make sense of how this structure relates to the graph.

Provide access to graphing technology. Consider arranging students in groups of 2. For the first two questions involving graphing, ask one partner to operate the graphing technology and the other to record the group’s observations, and then to switch roles halfway.

Consider pausing for a class discussion after the second question so students can share their observations. When discussing the effects of adding a constant term to \((x-h)^2\) , ask students how the vertical movement of the vertex affects the \(x\) -intercepts of the graph. If not mentioned by students, point out that in some cases it produces graphs that are away from the \(x\) -axis and thus have no \(x\) -intercepts.

Encourage students to put their graphing technology out of reach while they work on the third question.

• Using graphing technology, graph \(y=x^2\) . Then, add different numbers to \(x\) before it is squared (for example, \(y=(x+4)^2\) , \(y=(x-3)^2\) ) and observe how the graph changes. Record your observations.

Graph \(y=(x-1)^2\) . Then, experiment with each of the following changes to the function and see how they affect the graph and the vertex:

• Adding different constant terms to \((x-1)^2\) (for example: \((x-1)^2+5\) , \((x-1)^2-9\) ).
• Multiplying \((x-1)^2\) by different coefficients (for example: \(y=3(x-1)^2\) , \(y=\text-2(x-1)^2\) ).
• Use graphing technology to check your predictions. If they are incorrect, revise them. Then, complete the last row of the table.

Are you ready for more?

• What is the vertex of this graph?
• Find a quadratic equation whose graph has the same vertex and adjust it, if needed, so that it has the graph provided.

Consider displaying the incomplete table and inviting students to write in one of their correct responses. Then, ask students what they notice about the structure or composition of the expressions. Draw students’ attention to the following:

• All expressions in the table have a squared term.
• Most of the squared terms contain a sum or a difference of \(x\) and a number. For the one example where the squared term is not a sum or a difference, we can think of it as having a sum of \(x\) and 0.
• The squared term may have a coefficient, which can be positive or negative.
• Most expressions have a constant term. For the one example without a constant term, we can think of the constant term as 0.

To help students consolidate their observations, display the following sentence starters and ask students to complete them based on their work:

• When a quadratic equation is in vertex form of \(y = a(x-h)^2 + k\) , the coordinates of the vertex are \((\quad , \quad)\) .
• When the equation is graphed, the graph opens upward if . . .
• The graph opens downward if . . .

If not mentioned by students, point out that when a quadratic equation is in vertex form of \(y = a(x-h)^2 + k\) , the coordinates of the vertex are \((h,k)\) . Also point out that when the equation is graphed, the graph of the equation opens upward if \(a\) is positive and opens downward if \(a\) is negative.

Tell students that, in a future lesson, we will take a closer look at how the parts of a quadratic equation in vertex form,  \(y = a(x-h)^2 + k\) , produce the behaviors they observed on the graph. 

Lesson Synthesis

To help students see the connections between the three different forms of quadratic expressions and deepen their understanding of each, consider asking questions such as:

• “The standard form has a constant term: the \(c\) in \(ax^2+bx+c\) . The vertex form also has a constant term: the \(k\) in \(a(x-h)^2 +k\) . Are the two constant terms visible on the graph in the same way?” (No. The \(k\) in vertex form gives us the \(y\) -coordinate of the vertex. The \(c\) in standard form gives us the \(y\) -coordinate of the \(y\) -intercept. That said, changing these parameters has the same effect of moving the graph up or down.)
• “In both the vertex and standard form, the squared term \(x^2\) has a coefficient (which could be 1). Does this coefficient affect the graph similar ways?” (Yes, they both influence the direction of the opening of function’s graph and how wide or narrow the opening is.)

15.4: Cool-down - Visualizing A Graph (5 minutes)

Student lesson summary.

Sometimes the expressions that define quadratic functions are written in vertex form . For example, if the function \(f\) is defined by  \((x-3)^2 + 4\) , which is in vertex form, we can write  \(f(x)=(x-3)^2 + 4\)  and draw this graph to represent \(f\) .

The vertex form can tell us about the coordinates of the vertex of the graph of a quadratic function. The expression \((x-3)^2\) reveals that the vertex has \(x\) -coordinate 3, and the constant term of 4 reveals its \(y\) -coordinate. Here the vertex represents the minimum value of the function \(f\) , and its graph opens upward.

In general, a quadratic function expressed in vertex form is written as: \(\displaystyle y = a(x-h)^2 + k\) The vertex of its graph is at \((h,k)\) . The graph of the quadratic function opens upward when the coefficient  \(a\) is positive and opens downward when \(a\) is negative.

In future lessons, we will explore further how \(a\) , \(h\) , and \(k\) affect the graph of a quadratic function.

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About This Course

Welcome to the Math Medic Algebra 1 course! Here you will find a ready-to-be-taught lesson for every day of the school year, along with expert tips and questioning techniques to help the lesson be successful. Each lesson is designed to be taught in an Experience First, Formalize Later (EFFL) approach, in which students work in small groups on an engaging activity before the teacher formalizes the learning.

In Algebra 1, students build deep conceptual understanding around the topics of equivalence, solving equations, and the differences between linear, quadratic, and exponential functions. An emphasis is placed on describing how these functions change. This course was created using the Common Core State Standards as a guide. The standards taught in each Math Medic Algebra 1 lesson can be found here . The unit overviews and learning targets for the Math Medic Algebra 1 course can be found here .

Math Medic Help

How Does it Change?

Lesson Narrative

In this lesson, students contrast visual patterns that show quadratic relationships with those that show linear and exponential relationships. To analyze the patterns, students generate tables of values, write expressions, and create graphs. They also encounter the term quadratic expression and learn that a quadratic relationship can be written using an expression with a squared term.

Discerning and extending different patterns of change prompts students to look for and make use of structure (MP7). Generating tables of values and generalizing the relationships prompts students to express regularity in repeated reasoning (MP8).

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate tools to solve problems. Consider making technology available, in case requested.

Learning Goals

Teacher Facing

• Comprehend that a “quadratic relationship” can be expressed with a squared term.
• Describe (orally and in writing) a pattern of change associated with a quadratic relationship.
• Determine and explain (orally and in writing) whether a visual pattern represents a linear, exponential, or quadratic relationship.

Student Facing

• Let’s describe some patterns of change.

Learning Targets

• I can describe how a pattern is growing.
• I can tell whether a pattern is growing linearly, exponentially, or quadratically.
• I know a quadratic expression has a squared term.

CCSS Standards

• HSA-SSE.A.1
• HSA-SSE.B.3
• HSF-BF.A.1.a

Glossary Entries

A quadratic expression in \(x\) is one that is equivalent to an expression of the form  \(ax^2 + bx + c\) , where \(a\) , \(b\) , and \(c\) are constants and \(a \neq 0\) .

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Additional Resources