To compare the graph of [tex]\(y = \sec(x + 3) - 7\)[/tex] with the graph of [tex]\(y = \sec(x)\)[/tex], we need to consider the transformations applied to the function [tex]\(y = \sec(x)\)[/tex].
### Step-by-Step Solution:
1. Horizontal Shift:
- The term [tex]\(x + 3\)[/tex] inside the secant function indicates a horizontal shift.
- In general, [tex]\(y = \sec(x + a)\)[/tex] represents a horizontal shift of [tex]\(a\)[/tex] units to the left if [tex]\(a > 0\)[/tex], and [tex]\(a\)[/tex] units to the right if [tex]\(a < 0\)[/tex].
- Here, [tex]\(x + 3\)[/tex] means shifting the graph to the left by 3 units.
2. Vertical Shift:
- The [tex]\(-7\)[/tex] outside the secant function indicates a vertical shift.
- In general, [tex]\(y = \sec(x) - b\)[/tex] represents a vertical shift of [tex]\(b\)[/tex] units down if [tex]\(b > 0\)[/tex], and [tex]\(b\)[/tex] units up if [tex]\(b < 0\)[/tex].
- Here, [tex]\(-7\)[/tex] means shifting the graph downward by 7 units.
### Conclusion:
By combining these transformations, the graph of [tex]\(y = \sec(x + 3) - 7\)[/tex] is the graph of [tex]\(y = \sec(x)\)[/tex] shifted 3 units to the left and 7 units down.
Therefore, the correct answer is:
- It is the graph of [tex]\(y = \sec(x)\)[/tex] shifted 3 units left and 7 units down.
