Turning Points: Mastering Maxima and Minima with Your TI-84

Algebra 2 Grades High School 2:30 Video
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Lesson Description

Learn to identify and calculate relative maximum and minimum points (turning points) of polynomial functions using a TI-84 graphing calculator. This lesson covers the essential steps and interpretations.

Video Resource

Finding Relative Maximum and Relative Minimum (Turning Points) Using Ti84

Mario's Math Tutoring

Duration: 2:30
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Key Concepts

  • Relative Maximum (Local Maximum)
  • Relative Minimum (Local Minimum)
  • Graphing Calculator Usage (TI-84)
  • Left Bound and Right Bound
  • X and Y Coordinates of Turning Points

Learning Objectives

  • Students will be able to define and identify relative maximum and minimum points on a graph.
  • Students will be able to use a TI-84 graphing calculator to find the coordinates of relative maximum and minimum points of a polynomial function.
  • Students will be able to interpret the calculator's output, including scientific notation.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concepts of relative maximum and minimum. Explain that these are the 'turning points' of a graph where the function changes from increasing to decreasing (maximum) or decreasing to increasing (minimum).
  • Calculator Setup (5 mins)
    Guide students to input the example equation (X³ - 4X² + 4) into their TI-84 calculator using the 'Y=' function. Ensure students know how to access the graphing function and see the graph of the equation.
  • Finding Relative Maximum (10 mins)
    Walk students through the steps to find the relative maximum: 1. Press '2nd' then 'CALC' (trace) 2. Select '4: maximum' 3. Use the left arrow key to move the cursor slightly to the left of the maximum point (left bound). Press 'ENTER'. 4. Use the right arrow key to move the cursor slightly to the right of the maximum point (right bound). Press 'ENTER'. 5. Press 'ENTER' again for the guess. Explain how to interpret the coordinates (x, y) of the maximum point.
  • Finding Relative Minimum (10 mins)
    Repeat the process to find the relative minimum: 1. Press '2nd' then 'CALC' (trace) 2. Select '3: minimum' 3. Use the left arrow key to move the cursor slightly to the left of the minimum point (left bound). Press 'ENTER'. 4. Use the right arrow key to move the cursor slightly to the right of the minimum point (right bound). Press 'ENTER'. 5. Press 'ENTER' again for the guess. Explain how to interpret the coordinates (x, y) of the minimum point.
  • Interpreting Results and Scientific Notation (5 mins)
    Discuss how the calculator might display values in scientific notation (e.g., 8.2E-7). Explain that E-7 means the decimal point should be moved 7 places to the left, resulting in a very small number close to zero. Show other scientific notation examples.
  • Practice and Application (10 mins)
    Provide students with additional polynomial functions to graph and find the relative maximum and minimum points. Encourage them to work independently or in pairs.

Interactive Exercises

  • Graphing Challenge
    Give students different polynomial functions. Each student will graph their assigned function on the TI-84, then find and record the coordinates of all relative maximum and minimum points. Compare answers as a class and discuss any discrepancies.

Discussion Questions

  • Why is it important to select a left bound and a right bound when finding the maximum or minimum?
  • How does changing the equation affect the location of the maximum and minimum points?
  • Can a function have more than one relative maximum or minimum? What does this tell us about the function?

Skills Developed

  • Graphing Calculator Proficiency (TI-84)
  • Interpreting Graphs of Polynomial Functions
  • Problem-Solving
  • Critical Thinking

Multiple Choice Questions

Question 1:

What is a relative maximum also known as?

Correct Answer: Local Maximum

Question 2:

When using the TI-84 to find a maximum, what does 'left bound' refer to?

Correct Answer: A point to the left of the maximum

Question 3:

What function on the TI-84 do you use to calculate maximums and minimums?

Correct Answer: CALC

Question 4:

What does 'E-7' mean in the context of the TI-84 calculator output?

Correct Answer: The number is very small, close to zero

Question 5:

After setting the left and right bounds, what is the last step to find the max or min?

Correct Answer: Press GUESS then ENTER

Question 6:

A polynomial function changes from decreasing to increasing at a:

Correct Answer: Relative Minimum

Question 7:

What is the first step to do after the equation has been entered into the calculator?

Correct Answer: Graph the function

Question 8:

If the calculator displays x=2.6666667, what is a better way to approximate the value?

Correct Answer: 2 2/3

Question 9:

The relative maximum and minimum values on a graph are also referred to as the:

Correct Answer: Turning Points

Question 10:

What keys do you press to get to the calculate menu on the TI-84?

Correct Answer: 2nd then TRACE

Fill in the Blank Questions

Question 1:

A relative ________ is a point where the function changes from increasing to decreasing.

Correct Answer: maximum

Question 2:

The 'CALC' menu is accessed by pressing 2nd and the ________ button.

Correct Answer: TRACE

Question 3:

When finding the minimum, you need to select a ________ bound and a right bound.

Correct Answer: left

Question 4:

A polynomial function changes from decreasing to increasing at a relative ________.

Correct Answer: minimum

Question 5:

The high and low points on a curve are also called ________ ________.

Correct Answer: turning points

Question 6:

E-5 means that the number is smaller than zero.

Correct Answer: 5

Question 7:

On a graph, the lowest point in a particular section of the curve is called a ________.

Correct Answer: minimum

Question 8:

Before finding the maximum or minimum points, you must first ________ the equation on the calculator.

Correct Answer: graph

Question 9:

The x and y ________ of the maximum and minimum points are important.

Correct Answer: coordinates

Question 10:

On a graph, the highest point in a particular section of the curve is called a ________.

Correct Answer: maximum

Educational Standards

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. 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).

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