Question

A coordinate grid with a line passing through the points (negative 2, negative 1) and (2, 0)
Use the graph of the line to answer the questions.

What is an equation of the line in point-slope form?

How can the point-slope form be written in function notation?

Answer 1

To find the equation of the line that passes through the points (-2, -1) and (2, 0), we first need to calculate the slope (m) of the line. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Plugging in the coordinates of the points (-2, -1) and (2, 0):

\[
m = \frac{0 - (-1)}{2 - (-2)} = \frac{1}{4}
\]

Now that we have the slope \(m = \frac{1}{4}\), we can use either point to write the equation in point-slope form. The point-slope form of a line is given by:

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

Using the point (-2, -1):

\[
y - (-1) = \frac{1}{4}(x - (-2))
\]
\[
y + 1 = \frac{1}{4}(x + 2)
\]

That's the equation of the line in point-slope form.

Next, to write this in function notation, we can rearrange the equation into the form \(f(x)\):

\[
f(x) = \frac{1}{4}(x + 2) - 1
\]

Now, we simplify it:

\[
f(x) = \frac{1}{4}x + \frac{1}{2} - 1
\]
\[
f(x) = \frac{1}{4}x - \frac{1}{2}
\]

So, the final forms are:

1. Point-slope form: \(y + 1 = \frac{1}{4}(x + 2)\)
2. Function notation: \(f(x) = \frac{1}{4}x - \frac{1}{2}\)