Number of Solutions (from a graph)
Zeros of a function
We can write down the equation of a function in the form $y=f\left(x\right)$y=f(x). For example, let's consider the quadratic function with equation $y=x^2-1$y=x2−1. This equation has infinitely many solutions, such as the pair $x=2,y=3$x=2,y=3, with each $x$x-value corresponding to a $y$y-value.
Now, there is a particular set of solutions which have $y=0$y=0. We call the corresponding values of $x$x the zeros of the function. For some functions, such as $y=x^2-1$y=x2−1, we can set the $y$y-value to be $0$0 and solve the resulting equation to find the zeros. For other functions, such as $y=e^x-3x^2$y=ex−3x2, it is not so easy to find the zeros algebraically.
Finding the number of zeros graphically
In many situations it is useful to know how many zeros a function has, without necessarily finding the particular values. To do so, we can make use of the graph of the function.
Here is the graph of a function $y=f\left(x\right)$y=f(x):
The graph of a function
The zeros of a function are the values of $x$x which correspond to $y=0$y=0. Looking at the graph, we can see that all points which have a $y$y-coordinate of $0$0 lie on the $x$x-axis. That is, the zeros of a function are the $x$x-coordinates of the $x$x-intercepts of the curve!
For the function in the graph above we can see that the curve crosses the $x$x-axis three times, so this function has three zeros. The intercepts are highlighted in the following graph:
The graph of a function and its $x$x-intercepts
Note that not all functions have zeros! This can also be seen by looking at a graph of the function. If the curve does not intersect the $x$x-axis at any point, then there are no values of $x$x which correspond to $y=0$y=0 and so the function has no zeros. Here is the graph of $y=x^2+1$y=x2+1 as an example:
A graph of the function $y=x^2+1$y=x2+1, which has no $x$x-intercepts
For a function $y=f\left(x\right)$y=f(x), the zeros of that function are the values of $x$x which correspond to $y=0$y=0.
Graphically, the zeros of a function are the $x$x-coordinates of the $x$x-intercepts of the curve, and so the number of zeros of a function is the same as the number of $x$x-intercepts of its graph.
Not every function has zeros! For such functions, their graph does not cross the $x$x-axis at any point.
Finding the number of solutions to an equation
We can also use this technique when looking for the number of solutions to an equation, such as $x^3=x-3$x3=x−3.
To do so, we first rearrange the equation so that one side is equal to zero:
$x^3-x+3=0$x3−x+3=0
The solutions to this equation are the zeros of the function $y=x^3-x+3$y=x3−x+3. So we can determine the number of solutions by sketching a graph of $y=x^3-x+3$y=x3−x+3 (using technology or otherwise) and looking at the number of $x$x-intercepts:
A graph of the function $y=x^3-x+3$y=x3−x+3
Looking at the graph we can see that this curve has one $x$x-intercept, and so the original equation $x^3=x-3$x3=x−3 has exactly one real solution.
Practice questions
Question 1
A graph of the function $y=4x+3$y=4x+3 is shown below.
How many zeros does this function have?
Question 2
A graph of the function $y=\frac{1}{2}x^2-4x+8$y=12x2−4x+8 is shown below.
How many zeros does this function have?
Question 3
A graph of the function $y=\left(x-2\right)\left(x^2-6x+3\right)$y=(x−2)(x2−6x+3) is shown below.
How many values of $x$x satisfy the equation $\left(x-2\right)\left(x^2-6x+3\right)=0$(x−2)(x2−6x+3)=0?