To determine which points are on the graph of \( y = \tan x \), we need to evaluate the \( y \)-values for the corresponding \( x \)-coordinates and check if they match the given pairs.
-
\( \left( \frac{4\pi}{3}, \sqrt{3} \right) \): \[ \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \] This point is on the graph.
-
\( \left( \frac{\pi}{4}, -1 \right) \): \[ \tan\left(\frac{\pi}{4}\right) = 1 \] Since \( y \) should be \( 1 \), this point is not on the graph.
-
\( \left( \frac{\pi}{6}, -\frac{\sqrt{3}}{3} \right) \): \[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \quad \text{(This is incorrect; it should be positive.)} \] The point is not on the graph.
-
\( \left( \frac{\pi}{3}, \sqrt{3} \right) \): \[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \] This point is on the graph.
-
\( \left( 1, -\frac{\pi}{4} \right) \): The tangent of \( 1 \) (in radians) does not equal \( -\frac{\pi}{4} \). Therefore, this point is not on the graph.
Summary of Points on the Graph
The points that lie on the graph of \( y = \tan x \) are:
- \( \left( \frac{4\pi}{3}, \sqrt{3} \right) \)
- \( \left( \frac{\pi}{3}, \sqrt{3} \right) \)
So the correct answers are:
- \( \left( \frac{4\pi}{3}, \sqrt{3} \right) \)
- \( \left( \frac{\pi}{3}, \sqrt{3} \right) \)
1 answer