y = sec(x) and the point (3,2) are shown.  Graph y= - sec(3x) and the transformation of the point.

The graph of \( y = -\sec(3x) \) is a transformation of the graph of \( y = \sec(x) \). We can break this transformation into two main steps:

1. **Horizontal compression**: The factor of \( 3 \) inside the secant function compresses the graph horizontally by a factor of \( \frac{1}{3} \). This affects the period of the secant function, reducing it from \( 2\pi \) (for \( y = \sec(x) \)) to \( \frac{2\pi}{3} \) (for \( y = \sec(3x) \)).
   
2. **Vertical reflection**: The negative sign in front of the secant function reflects the graph over the x-axis.

### Transformation of the Point \( (3, 2) \):

To find the transformed point from \( (3, 2) \) on the graph of \( y = -\sec(3x) \), we apply the same transformations to the x and y coordinates.

- **Horizontal compression**: The x-coordinate is compressed by a factor of \( \frac{1}{3} \). Therefore, \( x = 3 \) becomes:
  \[
  x' = \frac{3}{3} = 1
  \]

- **Vertical reflection**: The y-coordinate is reflected across the x-axis, so \( y = 2 \) becomes:
  \[
  y' = -2
  \]

Thus, the transformed point is \( (1, -2) \).

### Final Transformed Graph
The graph of \( y = -\sec(3x) \) will appear as a compressed and flipped version of the original secant graph. The transformed point is \( (1, -2) \).

Would you like a graph of this transformation, further details on any steps, or explanations on secant function properties?

Here are 5 related questions:
1. How does the period of \( y = \sec(3x) \) compare to \( y = \sec(x) \)?
2. What are the asymptotes for \( y = \sec(x) \) and \( y = \sec(3x) \)?
3. How does a horizontal compression affect the shape of periodic functions?
4. What other transformations could you apply to \( y = \sec(x) \)?
5. How does the transformation of points change with horizontal or vertical shifts?

**Tip**: For transformations, always handle horizontal changes (inside the function) and vertical changes (outside the function) separately to avoid confusion!

y = sec(x) and the point (3,2) are shown. Graph y= - sec(3x) and the transformation of the point.

Explore the graph of y = -sec(3x) and learn how to transform the point (3,2) with horizontal compression and vertical reflection. This problem covers trigonometric transformations, including secant function properties and graphing techniques, suitable for Grades 10-12.

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