Question

look at this graph: what is the equation of the line in point - slope form? use the red point in your equation. write your answer using integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Identify the red point

The red point is at \((-20, -20)\) (from the graph: \(x = -20\), \(y = -20\)).

Step2: Find the slope

We can find another point on the line. Let's take the \(y\)-intercept? Wait, let's see the line. Let's take two points: when \(x = -100\), \(y = 0\)? Wait no, wait the line goes through, let's check the grid. Wait, from \((-100, 0)\) to \((0, -20)\)? Wait no, the red point is \((-20, -20)\). Wait, let's calculate slope between two points. Let's take \((-100, 0)\) and \((0, -20)\). Wait, no, maybe better to take \((-100, 0)\) and \((-20, -20)\). Wait, slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1, y_1)=(-100, 0)\) and \((x_2, y_2)=(-20, -20)\). Then \(m=\frac{-20 - 0}{-20 - (-100)}=\frac{-20}{80}=-\frac{1}{4}\). Wait, or another pair: \((-20, -20)\) and \((20, -30)\)? Wait no, maybe I made a mistake. Wait, let's check the graph again. The line is decreasing, so slope is negative. Let's see, from \(x=-100\) (left end) to \(x=0\), how much does \(y\) change? At \(x=-100\), \(y\) is 0? Wait, the left end is at \((-100, 0)\) (since the arrow is at \(-100\) on x-axis, and y=0). Then at \(x=0\), y is -20? Wait, no, the red point is at \((-20, -20)\). Wait, let's recalculate slope between \((-100, 0)\) and \((-20, -20)\):

\(m=\frac{-20 - 0}{-20 - (-100)}=\frac{-20}{80}=-\frac{1}{4}\). Yes, that's correct.

Step3: Point - slope form

Point - slope form is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the red point \((-20, -20)\) and \(m = -\frac{1}{4}\).

So substitute \(x_1=-20\), \(y_1=-20\), \(m = -\frac{1}{4}\) into the formula:

\(y - (-20)=-\frac{1}{4}(x - (-20))\)

Simplify the signs:

\(y + 20=-\frac{1}{4}(x + 20)\)

Answer:

\(y + 20 = -\frac{1}{4}(x + 20)\)