# Lesson 11

Zeros of Functions and Intercepts of Graphs

These materials, when encountered before Algebra 1, Unit 6, Lesson 11 support success in that lesson.

## 11.1: Which Output is 0? (10 minutes)

### Warm-up

The purpose of this activity is to give students a chance to practice some arithmetic with signed numbers, and remind them of the meaning of zero of a function. This is in preparation for the remainder of this lesson, when students will look for inputs to a function that give an output of 0.

### Launch

Arrange students in groups of 2. Give them a few minutes to quiet think time followed by a chance to share their thinking with a partner. Follow with whole-class discussion.

### Student Facing

Which of these functions have an output of 0 when the input is -4?

• $$v(x)=4x$$
• $$w(x)=\text-4x$$
• $$y(x)=8+2x$$
• $$z(x)=2x-8$$

### Activity Synthesis

Emphasize substituting the -4 and calculating the output rather than looking for patterns or shortcuts. Questions for discussion:

• “How do you know that the output of function $$y$$ is 0 when the input is -4?” (I started with $$y(x)=8+2x$$ and used -4 as the input, writing $$y(\text-4)=8+2(\text-4)$$. Evaluating this, I got $$8+\text-8$$, which is 0.
• “How do you know that the output of (each of the other functions) is not 0 when the input is -4?” (When I substituted a -4 in place of $$x$$ and evaluated the expression, the outcome was something other than 0.)
• “A term we’ve learned in this unit is zero of a function. What is an example of a zero of a function in this activity?” (-4 is a zero of function $$y$$, because $$y(\text-4)=0$$. However, -4 is not a zero of any of the other functions.)

Tell students that in the next activity, they are going to be hunting for zeros of some different functions.

## 11.2: Intercept Detective (15 minutes)

### Activity

The purpose of this activity is to strengthen connections between zeros of a function (input(s) when the output is 0), the output when the input is 0, and the coordinates of the intercept(s) of a graph of the function.

### Launch

To ensure that students understand the task before beginning, display one of the functions for all to see, and test an input value that is not a zero of the function. For example, evaluate $$a(\text-5)=\text-5 - 5$$. Is the output 0? No, the output is -10. So -5 is not an input that would give an output of 0. Draw students’ attention to the list of possible inputs, and tell them that the task is to find the input or inputs for each function that gives an output of 0.

### Student Facing

Here are the definitions of some functions, followed by some possible inputs for the functions.

$$a(x)=x - 5$$

$$b(x)=x + 5$$

$$c(x)=x-3$$

$$d(x)=x+1$$

$$f(x)=3x - 6$$

$$g(x)=3x + 6$$

$$h(x)=(x+5)(x+3)$$

$$m(x)=(x+1)(x-3)$$

$$n(x)=(3x-6)(x-5)$$

Possible inputs: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.

1. For each function, decide which input or inputs would give an output of 0.
2. Here are graphs of $$b$$, $$f$$, and $$m$$. Label each intercept with its coordinates, and be prepared to explain how you know.

### Activity Synthesis

Display the correct responses to the first question and ask students to check their work. Invite a student to demonstrate how they know that $$g(\text-2)=0$$ and $$m(3)=0$$ (or choose a different function that any students struggled with). The important takeaway is not to look for shortcuts or patterns, but rather understand by evaluating the function why a given number is a zero of the function.

Invite students to share their responses and reasoning to the second question. If not mentioned, emphasize that:

• The $$y$$-intercept of a graph corresponds to the output when the input is 0, so to find it, evaluate the function when $$x$$ is 0. For example, $$b(0)=5$$, so the coordinates are $$(0,5)$$ of an intercept of the graph.
• The $$x$$-intercepts correspond to the values found in the first question, the inputs that give an output of 0.
• Any time you are dealing with an intercept, one of the coordinates must be 0. The coordinates of an $$x$$-intercept are $$(\text{something},0)$$. The coordinates of a $$y$$-intercept are $$(0,\text{something})$$.

## 11.3: Making More Connections (20 minutes)

### Activity

This activity is an opportunity to practice making the connections discussed in the previous activity.

### Launch

If desired, arrange students in groups of 2, and ask them to take turns explaining each part of each question to each other. Make sure both partners agree before moving to the next part.

### Student Facing

1. For each function, identify the input that would give an output of 0.
• $$p(x) = x + 10$$
• $$q(x) = x - 10$$
• $$r(x) = 8 - x$$
• $$s(x) = \text-8 - x$$
• $$t(x) = 2x - 8$$
• $$u(x) = 2x + 8$$
2. Match each graph to a function in the previous question. Be prepared to explain your matches.
3. Label the intercepts on each graph with their coordinates.
4. For each function, identify the inputs that would give an output of 0.
• $$v(x) = (x + 10)(2x - 8)$$
• $$w(x) = (2x + 8)(10 - x)$$
5. Create three different functions whose output is 0 when the input is 7. At least one of your functions must be quadratic.

### Student Response

• “Which of these functions are quadratic? How do you know?” (The expression can be written using an $$x^2$$ term. It has two factors each containing an $$x$$.)
• “What can you say about the intercepts of the graphs of these functions?” (The graph of each function contains the point $$(7,0)$$. The $$y$$-intercept isn’t related to 7, because the $$y$$-intercept of the graph is the output of the function when the input is 0.)
The text passage contains multiple mathematical questions and activities, primarily focused on identifying the inputs that give an output of 0 for various functions and understanding the relationship between these inputs and the intercepts of the graphs of the functions.
multiple_questions_with_answers

1. **11.1: which output is 0?**
   - **question:** which of these functions have an output of 0 when the input is -4?
   - **answer:** the function \( y(x) = 8 + 2x \) has an output of 0 when the input is -4.

2. **11.2: intercept detective**
   - **question:** for each function, decide which input or inputs would give an output of 0.
   - **answer:** the inputs that give an output of 0 for each function are:
     - \( a(x) = x - 5 \): \( x = 5 \)
     - \( b(x) = x + 5 \): \( x = -5 \)
     - \( c(x) = x - 3 \): \( x = 3 \)
     - \( d(x) = x + 1 \): \( x = -1
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Lesson 11

Zeros of Functions and Intercepts of Graphs

These materials, when encountered before Algebra 1, Unit 6, Lesson 11 support success in that lesson.

## 11.1: Which Output is 0? (10 minutes)

### Warm-up

The purpose of this activity is to give students a chance to practice some arithmetic with signed numbers, and remind them of the meaning of zero of a function. This is in preparation for the remainder of this lesson, when students will look for inputs to a function that give an output of 0.

### Launch

Arrange students in groups of 2. Give them a few minutes to quiet think time followed by a chance to share their thinking with a partner. Follow with whole-class discussion.

### Student Facing

Which of these functions have an output of 0 when the input is -4?

• $$v(x)=4x$$
• $$w(x)=\text-4x$$
• $$y(x)=8+2x$$
• $$z(x)=2x-8$$

### Activity Synthesis

Emphasize substituting the -4 and calculating the output rather than looking for patterns or shortcuts. Questions for discussion:

• “How do you know that the output of function $$y$$ is 0 when the input is -4?” (I started with $$y(x)=8+2x$$ and used -4 as the input, writing $$y(\text-4)=8+2(\text-4)$$. Evaluating this, I got $$8+\text-8$$, which is 0.
• “How do you know that the output of (each of the other functions) is not 0 when the input is -4?” (When I substituted a -4 in place of $$x$$ and evaluated the expression, the outcome was something other than 0.)
• “A term we’ve learned in this unit is zero of a function. What is an example of a zero of a function in this activity?” (-4 is a zero of function $$y$$, because $$y(\text-4)=0$$. However, -4 is not a zero of any of the other functions.)

Tell students that in the next activity, they are going to be hunting for zeros of some different functions.

## 11.2: Intercept Detective (15 minutes)

### Activity

The purpose of this activity is to strengthen connections between zeros of a function (input(s) when the output is 0), the output when the input is 0, and the coordinates of the intercept(s) of a graph of the function.

### Launch

To ensure that students understand the task before beginning, display one of the functions for all to see, and test an input value that is not a zero of the function. For example, evaluate $$a(\text-5)=\text-5 - 5$$. Is the output 0? No, the output is -10. So -5 is not an input that would give an output of 0. Draw students’ attention to the list of possible inputs, and tell them that the task is to find the input or inputs for each function that gives an output of 0.

### Student Facing

Here are the definitions of some functions, followed by some possible inputs for the functions.

$$a(x)=x - 5$$

$$b(x)=x + 5$$

$$c(x)=x-3$$

$$d(x)=x+1$$

$$f(x)=3x - 6$$

$$g(x)=3x + 6$$

$$h(x)=(x+5)(x+3)$$

$$m(x)=(x+1)(x-3)$$

$$n(x)=(3x-6)(x-5)$$

Possible inputs: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.

1. For each function, decide which input or inputs would give an output of 0.
2. Here are graphs of $$b$$, $$f$$, and $$m$$. Label each intercept with its coordinates, and be prepared to explain how you know.

### Activity Synthesis

Display the correct responses to the first question and ask students to check their work. Invite a student to demonstrate how they know that $$g(\text-2)=0$$ and $$m(3)=0$$ (or choose a different function that any students struggled with). The important takeaway is not to look for shortcuts or patterns, but rather understand by evaluating the function why a given number is a zero of the function.

Invite students to share their responses and reasoning to the second question. If not mentioned, emphasize that:

• The $$y$$-intercept of a graph corresponds to the output when the input is 0, so to find it, evaluate the function when $$x$$ is 0. For example, $$b(0)=5$$, so the coordinates are $$(0,5)$$ of an intercept of the graph.
• The $$x$$-intercepts correspond to the values found in the first question, the inputs that give an output of 0.
• Any time you are dealing with an intercept, one of the coordinates must be 0. The coordinates of an $$x$$-intercept are $$(\text{something},0)$$. The coordinates of a $$y$$-intercept are $$(0,\text{something})$$.

## 11.3: Making More Connections (20 minutes)

### Activity

This activity is an opportunity to practice making the connections discussed in the previous activity.

### Launch

If desired, arrange students in groups of 2, and ask them to take turns explaining each part of each question to each other. Make sure both partners agree before moving to the next part.

### Student Facing

1. For each function, identify the input that would give an output of 0.
• $$p(x) = x + 10$$
• $$q(x) = x - 10$$
• $$r(x) = 8 - x$$
• $$s(x) = \text-8 - x$$
• $$t(x) = 2x - 8$$
• $$u(x) = 2x + 8$$
2. Match each graph to a function in the previous question. Be prepared to explain your matches.
3. Label the intercepts on each graph with their coordinates.
4. For each function, identify the inputs that would give an output of 0.
• $$v(x) = (x + 10)(2x - 8)$$
• $$w(x) = (2x + 8)(10 - x)$$
5. Create three different functions whose output is 0 when the input is 7. At least one of your functions must be quadratic.

### Student Response

• “Which of these functions are quadratic? How do you know?” (The expression can be written using an $$x^2$$ term. It has two factors each containing an $$x$$.)
• “What can you say about the intercepts of the graphs of these functions?” (The graph of each function contains the point $$(7,0)$$. The $$y$$-intercept isn’t related to 7, because the $$y$$-intercept of the graph is the output of the function when the input is 0.)
user
The text passage contains multiple mathematical questions and activities, primarily focused on identifying the inputs that give an output of 0 for various functions and understanding the relationship between these inputs and the intercepts of the graphs of the functions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

1. **11.1: Which Output is 0?**
   - **Question:** Which of these functions have an output of 0 when the input is -4?
   - **Answer:** The function \( y(x) = 8 + 2x \) has an output of 0 when the input is -4.

2. **11.2: Intercept Detective**
   - **Question:** For each function, decide which input or inputs would give an output of 0.
   - **Answer:** The inputs that give an output of 0 for each function are:
     - \( a(x) = x - 5 \): \( x = 5 \)
     - \( b(x) = x + 5 \): \( x = -5 \)
     - \( c(x) = x - 3 \): \( x = 3 \)
     - \( d(x) = x + 1 \): \( x = -1