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Algebra 2
Mastering Zeros: Finding X-Intercepts with Your TI-84

Mastering Zeros: Finding X-Intercepts with Your TI-84

Algebra 2 Grades High School 1:20 Video

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Lesson Description

Learn how to efficiently find the zeros (x-intercepts) of polynomial functions using the TI-84 graphing calculator. This lesson covers the essential steps and concepts to accurately identify and interpret zeros on a graph.

Video Resource

X-Intercepts (Zeros) How to Find Using Ti84

Mario's Math Tutoring

Duration: 1:20

Watch on YouTube

Key Concepts

Zeros of a function (x-intercepts)
Graphing polynomial functions on a TI-84 calculator
Using the 'zero' function on a TI-84 to find x-intercepts

Learning Objectives

Define the zeros (x-intercepts) of a function and explain their significance.
Input polynomial functions into a TI-84 graphing calculator.
Use the 'zero' function on a TI-84 to accurately find x-intercepts.
Interpret the x-intercepts in the context of the graph.

Educator Instructions

Introduction (5 mins)
Begin by defining zeros (x-intercepts) of a function. Explain that these are the points where the graph crosses the x-axis and the y-coordinate is zero. Briefly discuss the importance of finding zeros in various mathematical contexts.
TI-84 Setup and Input (5 mins)
Guide students on how to input a polynomial equation into the Y= editor on their TI-84 calculators. Use the example from the video (y = x^3 - 4x^2 + 4) as a starting point. Emphasize the importance of accurate input.
Graphing and Visual Identification (5 mins)
Instruct students to graph the function. Discuss how to visually identify the points where the graph crosses the x-axis. Highlight the limitations of visual estimation and the need for a more precise method.
Using the 'Zero' Function (10 mins)
Walk students through the steps of using the 'zero' function (found under 2nd -> CALC -> 2: zero). Explain the concepts of 'left bound,' 'right bound,' and 'guess.' Emphasize the importance of selecting a left bound to the left of the zero (above the x-axis) and a right bound to the right of the zero (below the x-axis). Demonstrate the process clearly.
Example Problems and Practice (10 mins)
Provide students with additional polynomial functions to graph and find the zeros using the TI-84. Encourage them to work independently or in pairs. Circulate to provide assistance and answer questions.

Interactive Exercises

Calculator Challenge
Provide students with a list of polynomial functions and challenge them to find all the real zeros of each function using their TI-84 calculators. Offer a small reward for the first student(s) to correctly identify all zeros for all functions.
Error Analysis
Intentionally input a function incorrectly or select inappropriate left/right bounds. Have students identify the error and explain how to correct it.

Discussion Questions

What does the zero of a function represent graphically?
Why is it important to choose appropriate left and right bounds when using the 'zero' function on the TI-84?
Can a polynomial function have multiple zeros? How can you identify them on the graph?
How can you use the TI-84 to confirm if you have found the correct zero?

Skills Developed

Using a TI-84 graphing calculator
Finding zeros (x-intercepts) of polynomial functions
Interpreting graphs of polynomial functions
Problem-solving using technology

Multiple Choice Questions

Question 1:

What is another term for the x-intercept of a function?

Y-intercept

Zero

Vertex

Asymptote

Correct Answer: Zero

Question 2:

On a graph, the zeros of a function are the points where the graph:

Crosses the y-axis

Touches the x-axis

Crosses or touches the x-axis

Reaches a maximum or minimum value

Correct Answer: Crosses or touches the x-axis

Question 3:

When using the 'zero' function on a TI-84, the 'left bound' should be:

To the right of the x-intercept

Above the x-axis and to the left of the x-intercept

Below the x-axis and to the left of the x-intercept

Anywhere on the graph

Correct Answer: Above the x-axis and to the left of the x-intercept

Question 4:

When using the 'zero' function on a TI-84, the 'right bound' should be:

To the left of the x-intercept

Above the x-axis and to the right of the x-intercept

Below the x-axis and to the right of the x-intercept

Anywhere on the graph

Correct Answer: Anywhere on the graph

Question 5:

What does the y-value equal at the zero of a function?

The function's maximum value

The function's minimum value

Correct Answer: 0

Question 6:

What is the first step in finding the zeros of a function on a TI-84 calculator?

Press the 'TRACE' button.

Enter the function into the 'Y=' editor.

Calculate the derivative of the function.

Find the vertex of the function.

Correct Answer: Enter the function into the 'Y=' editor.

Question 7:

After graphing a function on a TI-84, how do you access the 'zero' function?

Press 'STAT', then select 'CALC'.

Press '2nd', then 'TRACE' (CALC).

Press 'MATH', then select 'NUM'.

Press 'VARS', then select 'FUNCTION'.

Correct Answer: Press '2nd', then 'TRACE' (CALC).

Question 8:

What is the purpose of setting a 'guess' when using the 'zero' function on a TI-84?

To provide an initial estimate to help the calculator find the zero more quickly.

To manually calculate the zero.

To change the display settings of the graph.

The 'guess' function has no effect on the outcome.

Correct Answer: To provide an initial estimate to help the calculator find the zero more quickly.

Question 9:

If the TI-84 returns an error when using the 'zero' function, what is a likely cause?

The function has no real zeros.

The 'left bound' and 'right bound' are on the same side of the zero.

The calculator is in the wrong mode (e.g., radians instead of degrees).

All of the above.

Correct Answer: All of the above.

Question 10:

How can you verify that the zero found by the TI-84 is accurate?

By zooming in on the graph.

By substituting the zero value into the original function and checking if the result is close to zero.

By changing the window settings.

By finding the derivative at that point.

Correct Answer: By substituting the zero value into the original function and checking if the result is close to zero.

Fill in the Blank Questions

Question 1:

The x-intercepts of a graph are also known as the __________ of the function.

Correct Answer: zeros

Question 2:

To access the 'zero' function on the TI-84, you press '2nd' and then the __________ button.

Correct Answer: trace

Question 3:

The 'left bound' when finding a zero on the TI-84 should be to the __________ of the x-intercept.

Correct Answer: left

Question 4:

The y-coordinate of any x-intercept is always __________.

Correct Answer: zero

Question 5:

Before finding zeros, you must enter the function into the __________ editor on the TI-84.

Correct Answer: y=

Question 6:

The 'right bound' when finding a zero on the TI-84 should be to the __________ of the x-intercept.

Correct Answer: right

Question 7:

If a graph touches the x-axis but does not cross it, the point is still considered a __________.

Correct Answer: zero

Question 8:

The 'zero' function on a TI-84 is found under the __________ menu.

Correct Answer: calc

Question 9:

In the context of finding zeros of a function using a TI-84, the 'guess' prompt allows you to provide an __________ to the calculator.

Correct Answer: estimate

Question 10:

To ensure accuracy when using the 'zero' function, the left bound and right bound should be on __________ sides of the x-axis.

Correct Answer: opposite

Educational Standards

A2.A.1.4:Solve polynomial equations with real roots using various methods (e.g., polynomial division, synthetic division, using graphing calculators or other appropriate technology). A2.F.1.5:Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease.

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