Question

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)?

f(x) = –f(x) equals negative StartFraction 2 Over 3 EndFraction x minus 11.x – 11
f(x) = –f(x) equals negative StartFraction 2 Over 3 EndFraction x minus 1.x – 1
f(x) = –f(x) equals negative StartFraction 2 Over 3 EndFraction x plus 3.x + 3
f(x) = –f(x) equals negative StartFraction 2 Over 3 EndFraction x plus 13.x + 13

Answer 1

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:

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