Question

Which points are on the graph of y = tan x? Select all that apply.
(1 point)
Responses

left parenthesis Start Fraction 4 pi over 3 End Fraction comma Start Root 3 End Root right parenthesis
Image with alt text: left parenthesis Start Fraction 4 pi over 3 End Fraction comma Start Root 3 End Root right parenthesis

left parenthesis Start Fraction pi over 4 End Fraction comma negative 1 right parenthesis
Image with alt text: left parenthesis Start Fraction pi over 4 End Fraction comma negative 1 right parenthesis

left parenthesis Start Fraction pi over 6 End Fraction comma negative Start Fraction Start Root 3 End Root over 3 End Fraction right parenthesis
Image with alt text: left parenthesis Start Fraction pi over 6 End Fraction comma negative Start Fraction Start Root 3 End Root over 3 End Fraction right parenthesis

left parenthesis Start Fraction pi over 3 End Fraction comma Start Root 3 End Root right parenthesis
Image with alt text: left parenthesis Start Fraction pi over 3 End Fraction comma Start Root 3 End Root right parenthesis

left parenthesis 1 comma negative Start Fraction pi over 4 End Fraction right parenthesis
Image with alt text: left parenthesis 1 comma negative Start Fraction pi over 4 End Fraction right parenthesis

Answer 1

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.

1. **\( \left( \frac{4\pi}{3}, \sqrt{3} \right) \)**:
\[
\tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]
This point is on the graph.

2. **\( \left( \frac{\pi}{4}, -1 \right) \)**:
\[
\tan\left(\frac{\pi}{4}\right) = 1
\]
Since \( y \) should be \( 1 \), this point is not on the graph.

3. **\( \left( \frac{\pi}{6}, -\frac{\sqrt{3}}{3} \right) \)**:
\[
\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \quad \text{(This is incorrect; it should be positive.)}
\]
The point is not on the graph.

4. **\( \left( \frac{\pi}{3}, \sqrt{3} \right) \)**:
\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]
This point is on the graph.

5. **\( \left( 1, -\frac{\pi}{4} \right) \)**:
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:
- \( \left( \frac{4\pi}{3}, \sqrt{3} \right) \)
- \( \left( \frac{\pi}{3}, \sqrt{3} \right) \)