Finding Zeroes of Functions
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.
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.
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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