Zeros of a Function (Roots): Definition, TI 89 Steps

TI 89 Calculus > How to use the Zeros Function TI-89

What are Zeros of a Function?
What are Non Trivial Zeros?

What are Zeros of a Function?

The zeros (or roots) of a function are the point(s) where the input (x) produces a result (y) of zero.

zeros of a function ti 89
The roots here are labeled x1 and x2. This particular parabolic function has two roots or zeros.

If you plot a function on a graph, the roots are where the graph crosses the x-axis (i.e. where x = 0).

How to Find Roots of a Function

  1. Plot the function on a graph,
  2. Use algebra,
  3. TI-89 Instructions.

1. Plot the Function on a Graph

This is a fairly easy solution: plot your function and note where it crosses the x-axis. Caution: This will give you an approximate solution. For example, the graph might look like it crosses the x-axis at exactly x = 2. But it might actually cross at x = 1.9999 or x = 2.001. For an exact solution, use another method (like a graphing calculator).

Use algebra

This way requires you to have some strong algebra skills. There are multiple ways to solve using algebra, including the quadratic formula (see: curve sketching—roots for an example) and the rational number theorem.

Watch this short video to learn how to find the zeros of a function using the rational number theorem:

Finding All Zeros of a Polynomial Function Using The Rational Zero Theorem

Zeros of a Function on the TI 89 Steps

Use the Zeros Function on the TI-89 to find roots (or zeros) easily. The expression on the calculator is zeros(expression,var) where “expression” is your function and “var” is the variable you want to find zeros for (i.e. x or y variables).
Example problem 1: Find the zeros of this function: f(x) = x2 – 10x + 16

Step 1: Press the F2 key from the HOME screen and then press the number 4 button. This combination of buttons selects the “zeros” command.

Step 2: Press the following keys to enter the function into the command line: x ^ 2 – 1 0 x + 1 6, x )

Step 3: Press the ENTER key.

The resulting zeroes for this rational function will appear as a notation: ( 2 , 8 ) This means that the zeroes of this function are at x = 2 and x = 8.

That’s it! You’re done!

Tip: If you are asked to use the zeros function on the TI 89 to find zeros for a certain interval, set the interval using the “with” operator (a vertical slash |), the inequality operator (press the green diamond and then 0) and the “and” operator (+). For example, if you wanted to find the zeros of the equation sin(2x) – 3cos(x) between the interval 0 and 2π, the input would be:
zeros(sin(2x)-3cos(x),x)|0≤ and x≤2π

More Examples

Example problem 2: Find the roots of the following function graphically on the TI-89:
f(x) = x2 – 8x + 15

Step 1: Press the HOME key.

Step 2: Press the diamond (♦) key, then press F1 to enter into the y=editor. Press ENTER once to navigate to the input line at the bottom of the screen.

Step 3: Step 3: Press x ^ 2 – 8 x + 1 5 to enter the function into the “y1=” slot.

Step 4: Press the Enter key.

Step 5: Press the diamond (♦) key, then press F3 to view the graph of the function.

Step 6: Press the F5 key and then press 2 to select “Zero” (which is short for zeros of a function).

Step 7: Arrow to the left of the x-intercept for the “Lower Bound” and then press the ENTER key.

Step 8: Arrow to the right of the x-intercept for the “Upper Bound,” and then press the Enter key.

The TI-89 will return a value of 3 for “x” and 0 for “y” This means that one of the roots for the function is 3.

Step 9: Repeat steps 6 through 8. However, for the “Lower Bound,” arrow to a point that is to the right of the first intercept and then press the ENTER key. For the “Upper Bound,” arrow to the right of the x-intercept and press ENTER. The TI-89 will return a value of 5 for “x” and 0 for “y” This means that the other root for the function is 5.
That’s it!

Tip: Use the zoom function to see the graph more clearly on the screen. The zoom function is the F2 key when you are viewing the graph. Press 2 to Zoom In or press 3 to Zoom Out.

Roots of a function on the TI-89: Example 3

Example problem 3: Find the roots of the following function using the table feature on the TI-89:
f(x) = x2 – 7x + 12

Step 1: Press the Home key.

Step 2: Press the diamond key and then press F1 to enter into the y=editor.

Step 3:Press Enter to go down to the input line.

Step 4: Press x ^ 2 – 7 x + 1 2 to enter function into the “y1=” slot.

Step 5: Press Enter.

Step 6: Look to the left of the Y1 slot to make sure there is a check mark next to the function. If there isn’t a check mark, press the F4 key.

Step 7: Press the diamond key and then F5 to view the table of values for this function.

Step 7: Find what the variable “x” is equal to when “y1” is equal to zero on the table. Use the up and down scroll keys to scroll through the table. You should find that the x-values 3 and 4 satisfy this condition. These are the roots of this function.

That’s it! You’re done!

Tip: The table of values you generate will contain values for each checked function, so make sure you only have a check mark next to the function you want to draw a table for.

What are Non Trivial Zeros?

“Trivial” zeros are regular integers, while Non trivial zeros are roots that are complex numbers. The terms are unique to the Riemann Zeta Function.

They are referenced in the (famously unsolved) Riemann Hypothesis. Reimann calculated the first three. Since then, over 2 million have been found, but there is still no solution to the hypothesis.

Trivial and Non Trivial Zeros for the Zeta Function

For the Zeta function, the trivial zeros (i.e. the ones that are easy to define) are all negative even integers {−2, −4, −6,…}. The non trivial zeros are all others, and they are all complex numbers. Complex numbers have two parts: one real number (the ones you use in everyday life, like 1, 2, 3, …) and an imaginary part (i). The simplest imaginary number is the square root of zero (√0). If you haven’t come across these numbers before, you may want to check our overview of imaginary numbers and complex numbers.

More technically, non trivial zeros (denoted as ρ) happen for certain values of t, satisfying the equation:

s σ + i t

For s in the critical strip. The critical strip is defined as the region 0 < ρ < 1, where “ρ” is the real part of a complex number. All non trivial zeros of the Riemann Zeta function are in this strip.

Graphical Representation

The Riemann hypothesis is simply stated as “The real part of every non-trivial zero of the Riemann zeta function is ½”. With this in mind, the first six non trivial zeros (ρ1 to ρ6) are shown on the graph below:

non trivial zeros
“t” is the height of the zero.

The First Three Non Trivial Zeros

Accurate to 1,000 decimal places. Andrew Odlyzko has an extensive list of more than 2 million:





Andrew Odlyzko (n.d.).Tables of zeros of the Riemann zeta function. Retrieved November 25, 2019 from:
Brent, R. P. “On the Zeros of the Riemann Zeta Function in the Critical Strip.” Math. Comput. 33, 1361-1372, 1979.
Prentice Hall Calculator.

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