What are the 𝑥-intercepts of the function represented by the given graph?
In this question, we’re given the graph of a function, and we’re asked to determine the 𝑥-intercepts of the graph of this function. And we could first recall what we mean by the 𝑥-intercept of a function. It’s the points on the graph of the function where the 𝑦-coordinate is equal to zero. And the 𝑦-coordinates of these points are represented by the vertical axis. And this means the horizontal line 𝑦 is equal to zero contains all points with 𝑦-coordinate equal to zero.
Therefore, the 𝑥-intercepts of the function will be the points of intersection between the line 𝑦 is equal to zero and the graph of the function. This is why they’re called 𝑥-intercepts. The line 𝑦 is equal to zero is the 𝑥-axis. And we can see this curve intersects the 𝑥-axis at two points: when 𝑥 is negative one and when 𝑥 is three. And remember, the 𝑦-coordinates of these points will be zero since they lie on the 𝑥-axis. Therefore, we’ve shown the 𝑥-intercepts of the function represented by the graph are the points negative one, zero and three, zero.
And while this is enough to answer our question, it’s worth reiterating what is meant by the 𝑥-intercepts of a function. Why do we find these values? And what do they mean? Let’s start by giving our function a name. Let’s say it’s the function 𝑓 of 𝑥. In the graph of a function, the 𝑥-value of a point on the curve represents the input value of our function and the 𝑦-coordinate tells us the corresponding output of the function. For example, the vertex of this function is the point with coordinates one, four. When we input a value of one into our function, it outputs four. 𝑓 evaluated at one is four.
We can apply this exact same reasoning to the two 𝑥-intercepts of our function. This tells us 𝑓 evaluated at negative one is equal to zero, and 𝑓 evaluated at three is equal to zero. These are roots of the function. And the roots of functions tell us a lot of information about these functions, so they’re useful to find. Therefore, we were able to show the 𝑥-intercepts of the function represented by the given graph are the points negative one, zero and three, zero.