### Video Transcript

What are the ๐ฅ-intercepts of the function represented by the given graph?

In this question, weโre given the graph of a function, and weโre asked to determine the ๐ฅ-intercepts of the graph of this function. And we could first recall what we mean by the ๐ฅ-intercept of a function. Itโs the points on the graph of the function where the ๐ฆ-coordinate is equal to zero. And the ๐ฆ-coordinates of these points are represented by the vertical axis. And this means the horizontal line ๐ฆ is equal to zero contains all points with ๐ฆ-coordinate equal to zero.

Therefore, the ๐ฅ-intercepts of the function will be the points of intersection between the line ๐ฆ is equal to zero and the graph of the function. This is why theyโre called ๐ฅ-intercepts. The line ๐ฆ is equal to zero is the ๐ฅ-axis. And we can see this curve intersects the ๐ฅ-axis at two points: when ๐ฅ is negative one and when ๐ฅ is three. And remember, the ๐ฆ-coordinates of these points will be zero since they lie on the ๐ฅ-axis. Therefore, weโve shown the ๐ฅ-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, itโs worth reiterating what is meant by the ๐ฅ-intercepts of a function. Why do we find these values? And what do they mean? Letโs start by giving our function a name. Letโs say itโs the function ๐ of ๐ฅ. In the graph of a function, the ๐ฅ-value of a point on the curve represents the input value of our function and the ๐ฆ-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. ๐ evaluated at one is four.

We can apply this exact same reasoning to the two ๐ฅ-intercepts of our function. This tells us ๐ evaluated at negative one is equal to zero, and ๐ 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 theyโre useful to find. Therefore, we were able to show the ๐ฅ-intercepts of the function represented by the given graph are the points negative one, zero and three, zero.