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Which point does the graph of the parent function [tex][tex]$y = \tan(x)$[/tex][/tex] pass through?

A. [tex]\left(\frac{\sqrt{3}}{3}, \frac{\pi}{3}\right)[/tex]
B. [tex]\left(\frac{\pi}{3}, \frac{\sqrt{3}}{3}\right)[/tex]
C. [tex]\left(\frac{\pi}{3}, \sqrt{3}\right)[/tex]
D. [tex]\left(\sqrt{3}, \frac{\pi}{3}\right)[/tex]

Sagot :

GinnyAnswer
To determine which point the graph of the parent function [tex]\( y = \tan(x) \)[/tex] passes through, we need to evaluate the tangent function for specific values of [tex]\( x \)[/tex] and compare those results to the given points.

1. Consider the value [tex]\( x = \frac{\pi}{3} \)[/tex].
- Calculate [tex]\( y = \tan\left(\frac{\pi}{3}\right) \)[/tex].

The tangent of [tex]\( \frac{\pi}{3} \)[/tex] is a well-known trigonometric value:
[tex]\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \][/tex]

So, when [tex]\( x = \frac{\pi}{3} \)[/tex], the value of the function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \sqrt{3} \)[/tex]. This gives us the point [tex]\( \left(\frac{\pi}{3}, \sqrt{3}\right) \)[/tex].

Next, let's compare it with the given options:

- [tex]\(\left(\frac{\sqrt{3}}{3}, \frac{\pi}{3}\right)\)[/tex]: This point has the x-coordinate [tex]\(\frac{\sqrt{3}}{3}\)[/tex] and y-coordinate [tex]\(\frac{\pi}{3}\)[/tex], which does not match our point [tex]\(\left( \frac{\pi}{3}, \sqrt{3} \right) \)[/tex].

- [tex]\(\left(\frac{\pi}{3}, \frac{\sqrt{3}}{3}\right)\)[/tex]: This point has the y-coordinate [tex]\(\frac{\sqrt{3}}{3}\)[/tex], which is incorrect because our y-coordinate is [tex]\(\sqrt{3} \)[/tex], not [tex]\(\frac{\sqrt{3}}{3}\)[/tex].

- [tex]\(\left(\frac{\pi}{3}, \sqrt{3}\right)\)[/tex]: This point has the coordinates [tex]\(\left( \frac{\pi}{3}, \sqrt{3} \right)\)[/tex], which exactly matches our calculated point.

- [tex]\(\left(\sqrt{3}, \frac{\pi}{3}\right)\)[/tex]: In this case, the x-coordinate is [tex]\(\sqrt{3}\)[/tex] and the y-coordinate is [tex]\(\frac{\pi}{3}\)[/tex], which does not match our point [tex]\(\left( \frac{\pi}{3}, \sqrt{3} \right) \)[/tex].

Therefore, the correct point that the graph of the parent function [tex]\( y = \tan(x) \)[/tex] passes through is:

[tex]\[ \boxed{\left(\frac{\pi}{3}, \sqrt{3}\right)} \][/tex]
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