Finding Zeroes of Functions

Lesson Transcript
Instructor
David Karsner

David holds a Master of Arts in Education

Expert Contributor
Elaine Chan

Dr. Chan has a Ph.D. from the U. of California, Berkeley. She has done research and teaching in mathematics and physical sciences.

The zero of a function is the point (x, y) on which the graph of the function intersects with the x-axis. The y value of these points will always be equal to zero. There can be 0, 1, or more than one zero for a function. Updated: 08/09/2020

Finding the Zero

Let's assume that you like to paint landscapes and you have a lot of them in your attic. You're interested in making a little extra money so you thought you might sell them at the peddler's market. A booth at the market cost $465. You plan on selling your paintings for $15 a piece. A function is a process that takes one piece of data (the input) and then performs certain operations on the input and yields an output.

The function in this situation would be A = -465 + 15p, where A is the amount of money you make (the output), and p is the number of paintings you sell (the input). How many paintings would you need to sell to pay for the booth. In other words, where is the break-even point? This break-even point is the zero of this function. This lesson will show you how to find the zeros of several different kinds of functions as well as how to find them using a graphing calculator.

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  • 0:04 Finding the Zero
  • 1:02 What Is the Zero of a…
  • 1:51 Linear and Quadratic Functions
  • 2:56 Other Functions
  • 4:37 Using a Graphing Calculator
  • 5:53 Lesson Summary
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What Is the Zero of a Function?

The zero of a function is the x-value that when plugged into the function gives a y-value of zero. It goes by other names such as x-intercept and the root of the function. If given as an ordered pair, it will always have some number as the x-coordinate followed by a 0 for the y-coordinate. For example (4,0), (-2,0), and (0,0) could all be zeros of some function. Graphically the zero of a function is the intersection of the x-axis and the graph of the function. Different types of functions have different numbers of zeros. The graph of some functions does not cross the x-axis and therefore has no zeros (x-intercepts). Other functions have one or more. Finding these zeros is a very common task in algebra.

Linear and Quadratic Functions

Linear Functions are functions that can be put into the form y = mx + b. Their graphs are always lines. Linear functions will have at most one zero. The zero of a linear function can be found by replacing the y with zero and then solving for x.

Quadratic functions are functions that can be put in the form f(x) = ax2 + bx + c, which is called the standard form. Graphically these graphs are parabolas. The zeros of the function are where the f(x) = 0. These functions can have 0, 1, or 2 real zeros. There are several techniques for finding the zeros of a quadratic function, including the square root property, factoring, completing the square, and the quadratic formula. Of all these techniques, the quadratic formula is the most useful because it will work for all quadratic functions. It requires that you determine the values of a, b, and c, and then plug those values into the quadratic formula.

Other Functions

Let's look at a couple of the other functions that are out there.

1. Higher Order Polynomials

For polynomials that have a degree that is greater than 2, finding zeros becomes much more difficult. There is a slight possibility that the polynomial will factor. You can also use the rational root theorem, which says that IF a polynomial has a rational root (zero) it will exist at a value of x such that x is one of the factors of the constant term divided by one of the factors of the coefficient to the leading term. Notice that it was a big IF - many times the polynomial will not have a rational root. With higher order polynomials, the easiest method of finding the zero is the use of a graphing calculator.

2. Exponential and Logarithmic Functions

Exponential functions will be in the form of abx. If the exponential function fits this form and the value of the b is not zero; then the function will not have a zero. The graph will never cross the x-axis. The location of the y-intercept will be (0, a). Logarithmic functions are the inverse functions to exponential functions. If an exponential function has a y-intercept at (0, a), then its inverse logarithmic function will have a x-intercept (zero) at (a, 0).

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Additional Activities

Finding Approximate Zeroes of Functions


The formula for the approximate zero of f(x) is:

xn+1 = xn - f(xn ) / f'( xn ) .

Starting with n=1, you can get x2 .

Use x2 to get x3 and so on recursively.

In the limit as n goes to infinity, an infinite number of iterations, xn approaches the zero of the function.

This is a recursive formula that needs to be started with a reasonable initial guess. The function also needs to have a non-zero derivative. This method is called Newton's method or the Newton-Raphson method of root finding.


Question:


Use the Newton-Raphson method to find 21/6 .


Answer:


Finding 21/6 is equivalent to solving the equation

f(x) = x6 - 2 = 0


Taking the derivative of f(x):

f'(x) = 6x5


The recursion formula becomes:

xn+1 = xn - ( xn6 - 2) / 6xn 5 .

Using an initial guess of x1 = 1, we can generate the sequence of approximate roots as

1., 1.16666667, 1.12644368, 1.12249707, 1.12246205, 1.12246205, ...

Since the last two approximants agree to within eight decimal places we say that the last approximant is accurate to that extent.

21/6 is approximately 1.12246205

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