How to Find Maximum and Minimum Values on a TI-84 Calculator

Introduction

Finding maximum and minimum values is a fundamental skill in algebra, precalculus, calculus, physics, and economics. Whether you are finding the peak of a projectile, optimizing revenue in a business problem, or analyzing polynomial curves, your calculator can locate these critical points in seconds.

The TI-84 series features a dedicated CALC menu that automatically calculates local extrema directly from your plotted function. If you don’t have a physical device on hand, you can follow along directly using our free TI-84 Calculator Online in your browser.

When Should You Find Maximum and Minimum Values?

You will frequently need to calculate maximums and minimums in the following scenarios:

• Optimization Problems: Maximizing volume, surface area, or profit while minimizing material costs or time.
• Quadratic Analysis: Finding the exact vertex of upward or downward opening parabolas.
• Calculus Turning Points: Identifying relative extrema where the derivative (f'(x) = 0).
• Physics Trajectories: Determining maximum height reached by a projectile and time elapsed.

What Are Maximum and Minimum Values?

In graphing and calculus:

• Relative Maximum: The highest point a function reaches within a local open interval (the peak).
• Relative Minimum: The lowest point a function reaches within a local open interval (the valley).

For example, in the parabola (y = x^2 – 4x + 3), the curve opens upward with a vertex at ((2, -1)), representing a minimum value of (y = -1) at (x = 2).

Step-by-Step Instructions: Finding a Maximum Value

Follow these steps to find the peak (maximum) of a function such as (y = -x^2 + 4x + 5):

1. Open the Y= Editor: Press [Y=] and clear any existing functions.
2. Enter Your Function: Type (-) [X,T,θ,n] [x²] [+] 4 [X,T,θ,n] [+] 5.
3. Graph the Curve: Press [GRAPH]. If the curve is cut off, press [ZOOM] > 6:ZStandard.
4. Open the Calculate Menu: Press [2ND] followed by [TRACE] (which accesses [CALC]).
5. Select Maximum: Scroll down to 4:maximum and press [ENTER].
6. Set Left Bound: Use the left arrow key [◄] to move the blinking cursor slightly to the left of the peak, then press [ENTER]. A small marker arrow appears at the top.
7. Set Right Bound: Use the right arrow key [►] to move the cursor to the right of the peak, then press [ENTER].
8. Provide a Guess: Move the cursor close to the peak and press [ENTER].

Result: The calculator will scan the bounded region and display X = 2 and Y = 9 at the bottom of the screen. The maximum value of the function is 9.

Step-by-Step Instructions: Finding a Minimum Value

To find the lowest point of a curve (such as (y = x^2 – 4x + 3)):

1. Enter the Function: Press [Y=] and enter X² - 4X + 3.
2. Graph: Press [GRAPH] to view the parabola.
3. Open CALC Menu: Press [2ND] > [TRACE].
4. Select Minimum: Choose 3:minimum and press [ENTER].
5. Set Bounds & Guess: Set the Left Bound to the left of the vertex, the Right Bound to the right, and press [ENTER] on your Guess.

Result: The screen displays X = 2, Y = -1. The minimum value is -1.

Worked Examples

Worked Example 1: Finding a Profit Maximum

Problem: A business models its profit using the equation (P(x) = -x^2 + 6x – 2), where (x) is units produced in thousands. Find the maximum profit.

• Enter -X² + 6X - 2 into Y1 and press [GRAPH].
• Press [2ND] > [TRACE] > 4:maximum.
• Set Left Bound at (x = 2), Right Bound at (x = 4), and press [ENTER] on Guess.
• Solution: The calculator outputs (X = 3, Y = 7). Maximum profit is $7,000 at 3,000 units.

Worked Example 2: Finding a Minimum Cost

Problem: Find the minimum value of (C(x) = x^2 – 8x + 10).

• Enter X² - 8X + 10 into Y1 and press [GRAPH].
• Press [2ND] > [TRACE] > 3:minimum.
• Set Left Bound at (x = 3), Right Bound at (x = 5), and Guess at (x = 4).
• Solution: The calculator outputs (X = 4, Y = -6). The minimum value is -6.

Common Errors & Troubleshooting

• ERR: NO SIGN CHG: Occurs if your Left Bound and Right Bound do not trap the peak or valley between them. Always ensure the highest or lowest point sits visibly between both arrows.
• Peak Outside Window: If you cannot see the vertex, press [WINDOW] and expand Ymax or Ymin, or use [ZOOM] > 0:ZoomFit.
• Minus Sign Confusion: Remember to use the negative key [(-)] for leading negatives (e.g. (-x^2)), not the subtraction key [-].

Frequently Asked Questions

Can I find extrema on multiple graphed lines?

Yes. If you have multiple functions in Y=, use the up [▲] and down [▼] arrow keys to select which function you are analyzing before setting bounds.

How does this compare to finding derivatives in Calculus?

In calculus, extrema occur where (f'(x) = 0). The TI-84 numerical solver uses approximation algorithms to reach the exact same coordinate without requiring manual algebraic differentiation. You can also explore our guide on TI-84 Functions for Calculus Students for more advanced tools.

Summary

Finding maximums and minimums on a TI-84 is fast and reliable with the CALC menu ([2ND] + [TRACE]). By setting accurate left and right bounds, you can solve algebra and calculus optimization problems with complete confidence. Test your skills now on the live TI-84 simulator!