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.
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.
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).
3. Rational Functions
Finally, rational functions are functions in the form of f(x) = p(x) / q(x), where p(x) and q(x) are polynomials and q(x) can't equal zero. To find the zero of rational functions, find the zeros of p(x).
Using a Graphing Calculator
The graphing calculator can be used to find the real roots of functions. To find the zeros of a function with a graphing calculator, follow these steps. The directions given here are for the TI-83 and 84 brand of graphing calculators. Others brands will perform the same operations with similar buttons.
- Type the function into the y = function bank (top left button).
- Graph the function using the ''Graph'' button (top right button).
- Use the standard 10-by-10 window selecting the ''Zoom'' button and then z-standard.
- If you do not see an intersection of the graph and the x-axis, make your x-min and x-max values larger until you do.
- Once in sight, select the 2nd Calc.
- Select zero.
- The screen will say ''Left Bound;'' move the cursor to the left of the zero and hit ''Enter.''
- The screen will say ''Right Bound;'' move the cursor to the right of the zero and hit ''Enter.''
- The screen will say ''Guess;'' hit ''Enter.''
- Your x-value is your zero or root.
The graphing calculator will not find non-real roots.
All right, let's take a moment to review what we've learned about finding the zeros of a function, with a function being a process that takes one piece of data (the input) and then performs certain operations on the input and yields an output. As we learned, finding the zero of a function means to find the point (a, 0) where the graph of the function and the y-intercept intersect. To find the value of a from the point (a, 0), you need to set the function equal to zero and then solve for x. This involves using different techniques depending on the type of function that you have. The functions we looked at are as follows:
- Linear functions, which are functions that can be put into the form y = mx + b.
- Quadratic functions, which are functions that can be put in the form f(x) = ax2 + bx + c, which is called the standard form.
- Higher order polynomials, but using the rational root theorem, which says that IF a polynomial has a rational root (zero), then 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.
- Exponential functions, which is in the form of abx.
- Logarithmic functions, which are the inverse functions to exponential functions.
- Rational functions, which are functions in the form of f(x) = p(x) / q(x), where p(x) and q(x) are polynomials and q(x) can't equal zero.
We also learned how you can find the zero of a function by graphing the function on a graphing calculator and then searching for the point of intersection.
To unlock this lesson you must be a Study.com Member.
Create your account
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.
Use the Newton-Raphson method to find 21/6 .
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
Register to view this lesson
Unlock Your Education
See for yourself why 30 million people use Study.com
Become a Study.com member and start learning now.Become a Member
Already a member? Log InBack