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Unit 4, Lesson 10

Graphs of Functions in Standard and Factored Forms

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Integrated Math 2

10.1

Warm-up

5 mins

A Linear Equation and Its Graph

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

In this Warm-up, students revisit the graph of a linear equation and recall that an equation can tell us something about the graph that represents it and vice versa.

Launch

None

Activity

None

Student Task Statement

Here is a graph of the equation .

<p><strong>Graph of line, origin O. Horizontal x axis from negative 2 to 5 by 1’s. Vertical y axis from negative 2 to 12, by 2’s. Line starts at 0 comma 8, passes through 1 comma 6, 2 comma 4, and 4 comma 0.</strong></p>

Where do you see the 8 from the equation in the graph?
Where do you see the -2 from the equation in the graph?
What is the -intercept of the graph? How does this relate to the equation?

Student Response

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Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Make sure students recall that:

The -intercept tells us where a graph intersects the -axis, at which point the value is 0. When a linear equation is written in the form , the tells us the -intercept because when is 0, is also 0, so and is the -intercept.
The -intercept tells us where a graph intersects the -axis, at which point the value is 0. To find where the graph intersects the -axis, find the solution to .

Tell students that, when written in certain forms, quadratic expressions also tell us information about their graphs, and vice versa. We’ll explore these connections in this lesson and upcoming ones.

10.2

Activity

15 mins

Revisiting Projectile Motion

Instructional Routines

None

Materials

None

Activity Narrative

In the past few lessons, students have reasoned symbolically and formally about quadratic expressions. This activity ties that work back to the quadratic functions that represent situations from earlier in the unit.

Students are given two expressions that represent a familiar quadratic function. They identify these expressions by their form and justify why the two are equivalent, using what they learned about expanding factored expressions. Students also study the graph that represents the function and interpret the features of the graph in context. Students are asked to interpret the meaning of the horizontal and vertical intercepts rather than the - and -intercepts because these variables are not used to define the function.

10.3

Activity

15 mins

Relating Expressions and Their Graphs

Instructional Routines

None

Materials

None

Activity Narrative

The goal of this activity is to uncover the connections between the - and -intercepts on the graph and the parameters of quadratic expressions in standard and factored form. Earlier in the unit, students learned that the zeros of a quadratic function are -values that produce a -value of 0, and that the zeros of a function are the -coordinates of the -intercepts of the graph. This idea comes into focus in this activity.

Note that in the examples given here, the -coordinate of the -intercept is always equal to the product of the zeros, but this is not the case with all graphs representing quadratic functions. For example, the graph representing intercepts the -axis at 2 and -2, but the -intercept is . If students make this observation, consider acknowledging that this seems to be true for these graphs and consider prompting students to revisit this observation in upcoming lessons (in which students will also study graphs) to see if it is always true.

At this point, students are just noticing that numbers in the factored expressions have something to do with intercepts on the graph. In a later lesson students will explore why they are related.

Lesson Synthesis

In this lesson, we saw that quadratic expressions can give us clues about the intercepts of their graphs, and graphs can give us insights about the expressions they represent. Ask students:

“If we graph and , will we end up with the same graph? How do you know?” (Yes. The two expressions that define are equivalent. Expanding gives , or .)
“Where do you predict the intercepts of the graph will be? How do you make your predictions?” (The -intercept will be and the -intercepts will be and . In the examples in the lesson, we saw that the numbers in the factored form give a clue about the -intercepts, and the constant term in the standard form gives a clue about the -intercept. Also, evaluating when gives us the -intercept.)
“What do the -intercepts of a graph tell us about the quadratic function it represents?” (They tell us the zeros of the function.)

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Student Lesson Summary

Different forms of quadratic functions can tell us interesting information about the function’s graph. When a quadratic function is expressed in standard form, it can tell us the -intercept of the graph that represents the function.

For example, the graph representing has its -intercept at . This makes sense because the -coordinate is the -value when is 0. Evaluating the expression at gives , which equals 7.

<p><strong>Graph of non linear function, origin O. Horizontal axis from negative 2 to 5, by 1’s. Vertical axis from negative 2 to 8, by 2’s. Point plotted at 0 comma 7. Line decreases until about 2 point 5 comma 1, then increases.</strong></p>

When a function is expressed in factored form, it can help us see the -intercepts of its graph. Let’s look at function , given by and function , given by .

If we graph , we see that the -intercepts of the graph are and . Notice that 1 and 4 also appear in , and they are subtracted from .

If we graph , we see that the -intercepts are at and . Notice that 2 and 6 are also in the equation , but they are added to .

The connection between the factored form and the -intercepts of the graph tells us about the zeros of the function (the input values that produce an output of 0).

Launch

Arrange students in groups of 2. Give students a few minutes of quiet time to complete the first two questions. Then, ask them to discuss their observations with a partner, before completing the last question.

Activity

None

Student Task Statement

Here are pairs of expressions in standard and factored forms. Each pair of expressions define the same quadratic function, which can be represented with the given graph.

Identify the -intercepts and the -intercept of each graph.

Function

Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 4 to 4, by 2’s. Line passes through negative 4 comma 3, negative 3 comma 0, negative 2 comma negative 1, negative 1 comma 0, and 0 comma 3.

-intercepts:

-intercept:

Function

Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 4 to 4, by 2’s. Line passes through 0 comma 4, 1 comma 0, 2 comma negative 2, 3 comma negative 2, 4 comma 0, and 5 comma 4.

-intercepts:

-intercept:

Function

Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 10 to 2, by 2’s. Line passes through negative 3 comma 0, negative 2 comma negative 5, 0 comma negative 9, 1 comma negative 8, and 3 comma 0.

-intercepts:

-intercept:

Function

Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 8 to 2, by 2’s. Line passes through 0 comma 0, 1 comma negative 4, 2 comma negative 6, 3 comma negative 6, 4 comma negative 4, and 5 comma 0.

-intercepts:

-intercept:

Function

Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 2 to 6, by 2’s. Line passes through 0 comma 0, 1 comma 4, 2 comma 6, 3 comma 6, 4 comma 4, and 5 comma 0.

-intercepts:

-intercept:

Function

Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 2 to 6, by 2’s. Line passes through negative 4 comma 4, negative 2 comma 0, and 0 comma 4

-intercepts:

-intercept:
What do you notice about the -intercepts, the -intercept, and the numbers in the expressions defining each function? Make a couple of observations.
Here is an expression that models function , another quadratic function: . Predict the -intercepts and the -intercept of the graph that represent this function.

Student Response

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Audience

Assessment Access

Lesson 10

Graphs of Functions in Standard and Factored Forms

Warm-up

A Linear Equation and Its Graph

Standards Alignment

Building On

HSF-IF.B.4

Addressing

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Revisiting Projectile Motion

Instructional Routines

Materials

Activity Narrative

Activity

Relating Expressions and Their Graphs

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

Addressing

Building Toward

HSA-SSE.B.3

Standards Alignment

Building On

Addressing

HSA-SSE.B.3

Building Toward