Which linear function represents the line given by the point-slope equation y + 7 = –y plus 7 equals negative StartFraction 2 Over 3 EndFraction left-parenthesis x plus 6 right-parenthesis.(x + 6)?

To find the linear function that represents the line given by the point-slope form \( y + 7 = -\frac{2}{3}(x + 6) \), we first simplify and convert this equation into slope-intercept form (i.e., \( y = mx + b \)).

Starting with the point-slope equation:

\[ y + 7 = -\frac{2}{3}(x + 6) \]

Next, we distribute \( -\frac{2}{3} \):

\[ y + 7 = -\frac{2}{3}x - \frac{2}{3} \cdot 6 \] \[ y + 7 = -\frac{2}{3}x - 4 \]

Now, we isolate \( y \) by subtracting 7 from both sides:

\[ y = -\frac{2}{3}x - 4 - 7 \] \[ y = -\frac{2}{3}x - 11 \]

This can be rewritten in function notation:

\[ f(x) = -\frac{2}{3}x - 11 \]

Now, we compare this function to the options provided:

1. \( f(x) = -\frac{2}{3}x - 11 \)
2. \( f(x) = -\frac{2}{3}x - 1 \)
3. \( f(x) = -\frac{2}{3}x + 3 \)
4. \( f(x) = -\frac{2}{3}x + 13 \)

The function we derived, \( f(x) = -\frac{2}{3}x - 11 \), matches the first option.

Thus, the correct linear function is:

\( f(x) = -\frac{2}{3}x - 11 \).