Question

Which linear function equation below would contain the points (-6,-8) and (12,4)
Options
f(x)= 2/3x+ 4
f(x)= -2/3x+4
f(x) = -2/3x -4
f(x)= 2/3x -4

Answer 1

To determine which linear function contains the points \((-6, -8)\) and \((12, 4)\), we first need to find the slope of the line that goes through those points.

The formula for finding the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Plugging in the coordinates of the points:

- \( (x_1, y_1) = (-6, -8) \)
- \( (x_2, y_2) = (12, 4) \)

We calculate the slope \( m \):

\[
m = \frac{4 - (-8)}{12 - (-6)} = \frac{4 + 8}{12 + 6} = \frac{12}{18} = \frac{2}{3}
\]

Now that we have the slope, we can use the point-slope form of the equation of a line, which is:

\[
y - y_1 = m(x - x_1)
\]

Using one of the points, say \((-6, -8)\), our equation becomes:

\[
y - (-8) = \frac{2}{3}(x - (-6))
\]

Simplifying this:

\[
y + 8 = \frac{2}{3}(x + 6)
\]

Next, we distribute:

\[
y + 8 = \frac{2}{3}x + \frac{2}{3} \cdot 6
\]
\[
y + 8 = \frac{2}{3}x + 4
\]

Now, subtracting 8 from both sides:

\[
y = \frac{2}{3}x + 4 - 8
\]
\[
y = \frac{2}{3}x - 4
\]

Thus, the linear function that contains the points \((-6, -8)\) and \((12, 4)\) is:

\[
f(x) = \frac{2}{3}x - 4
\]

From the options given, the correct answer is:

**f(x) = 2/3x - 4**