Question Video: Finding ๐‘ฅ-Intercepts from a Given Graph Mathematics

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

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Video Transcript

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

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

Therefore, the ๐‘ฅ-intercepts of the function will be the points of intersection between the line ๐‘ฆ is equal to zero and the graph of the function. This is why theyโ€™re called ๐‘ฅ-intercepts. The line ๐‘ฆ is equal to zero is the ๐‘ฅ-axis. And we can see this curve intersects the ๐‘ฅ-axis at two points: when ๐‘ฅ is negative one and when ๐‘ฅ is three. And remember, the ๐‘ฆ-coordinates of these points will be zero since they lie on the ๐‘ฅ-axis. Therefore, weโ€™ve shown the ๐‘ฅ-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, itโ€™s worth reiterating what is meant by the ๐‘ฅ-intercepts of a function. Why do we find these values? And what do they mean? Letโ€™s start by giving our function a name. Letโ€™s say itโ€™s the function ๐‘“ of ๐‘ฅ. In the graph of a function, the ๐‘ฅ-value of a point on the curve represents the input value of our function and the ๐‘ฆ-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 ๐‘ฅ-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 theyโ€™re useful to find. Therefore, we were able to show the ๐‘ฅ-intercepts of the function represented by the given graph are the points negative one, zero and three, zero.

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