Question

What is the function rule for the line?



Responses

f(x)=−23x−2
f begin argument x end argument equals negative fraction 2 over 3 end fraction x minus 2.

f(x)=23x−2
f begin argument x end argument equals fraction 2 over 3 end fraction x minus 2.

f(x)=32x−2
f begin argument x end argument equals fraction 3 over 2 end fraction x minus 2.

f(x)=32x+2
f begin argument x end argument equals fraction 3 over 2 end fraction x plus 2.
A coordinate grid with x and y axis ranging from negative five to five. The line passes the points begin ordered pair 0 comma negative 2 end ordered pair and begin ordered pair 3 comma 0 end ordered pair.

Answer 1

To find the function rule for the line that passes through the points \((0, -2)\) and \((3, 0)\), we can first determine the slope of the line using the formula for the slope \(m\):

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

Using the points \((x_1, y_1) = (0, -2)\) and \((x_2, y_2) = (3, 0)\):

\[
m = \frac{0 - (-2)}{3 - 0} = \frac{2}{3}
\]

Now that we have the slope \(m = \frac{2}{3}\), we can use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), and use the point \((0, -2)\):

\[
y - (-2) = \frac{2}{3}(x - 0)
\]

This simplifies to:

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

Subtracting 2 from both sides gives:

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

Thus, the function rule for the line is:

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

So the correct answer from the provided options is:

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