Theodore O. answered • 02/22/20
A master of visualizing Mathematical and Grammatical Relationships
Quadratic functions can have one, two, or zero zeros. Zeros are also called the roots of quadratic functions, and they refer to the points where the function intersects the X-axis.
The standard form of a quadratic function is ax² + bx + c = 0. This form is specifically for determining what values of x will result in y equaling 0. However, we just want to know how many values of x result in y equaling 0.
So, we'll set our function to equal all values of y on our graphs. Recall that the graphs of quadratic functions always form parabolas. If we set B and C equal to zero and A to one, then we have the simplest quadratic function possible: y = x^2.
We see that it intersects the X-axis at only one point, the origin (0,0). Note that A is positive 1, so the parabola rises upward. Let's compare it with the function y = x^2 - 1. This parabola is one unit lower and crosses the X-axis at two points; it has two zeros. If we change C from -1 to plus 1, the parabola is completely above the X-axis, so it has no zeros. If we change the value of A from positive 1 to -1, the parabola will open downward, and it will have two zeros.
The B value can also change the number of zeros. Let's change it from 0 to -1 in our first function. Compare y = x^2 with y = x^2 - x. one zero vs. two zeros.
A graphing calculator will immediately reveal the number of zeros in a quadratic function, but if you are asked this question on an exam, you most likely won't be allowed to use a graphing calculator to answer it.
Fortunately, there's another simple method to determine the number of zeros. We use the quadratic formula. It can tell us what the values of the zeros are for a function, but we just want to know how many zeros there are, so we only need to use the square root part of the formula.
This expression under the square root symbol, b squared minus 4ac is called the discriminant. When the discriminant is positive, it will have both a positive and negative square root. As indicated by the plus or minus sign, this will result in two zeros. When the discriminant equals 0, there will be only one zero, and when it's negative, there will be no zeros.
Let's plug the A,B, and C values from our example functions into the discriminant to compare the results. For y = x^2, we have 0 squared - 4 times 1 times 0 or 0 minus 0. The square root of zero is zero, so this confirms that this function will have only one value of x that results in y equaling zero.
For y = x^2 - 1, the discriminant equals 0 squared - 4 times 1 times -1 or 0 plus 4. The square root of 4 is both plus 2 and minus 2, so that's why this function has two zeros.
For y = x^2 + 1, the discriminant equals 0 squared - 4 times 1 times 1 or 0 minus 4. The square root of -4 is undefined, so this function has no zeros.
When you encounter this type of question on an exam or in your homework, the values of A,B, and C typically won't be as simple as my example, but the method for solving the problem is exactly the same.
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