Question

what is the equation of the trend line in the scatter plot? use the two yellow points to write the equation in slope - intercept form. use coefficients as integers, proper fractions, or improper fractions in simplest form.

Explanation:

Step1: Identify the two yellow points

The yellow points are at \((0, 9)\) (when \(x = 0\), \(y = 9\)) and \((9, 1)\) (when \(x = 9\), \(y = 1\)).

Step2: Calculate the slope (\(m\))

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0,9)\) and \((x_2,y_2)=(9,1)\). Then \(m=\frac{1 - 9}{9 - 0}=\frac{-8}{9}=-\frac{8}{9}\)? Wait, no, wait. Wait, when \(x = 0\), \(y = 9\) (y - intercept, \(b = 9\)). When \(x = 9\), \(y = 1\). So the change in \(y\) is \(1 - 9=-8\), change in \(x\) is \(9 - 0 = 9\). Wait, but maybe I made a mistake. Wait, let's check the graph again. Wait, the first yellow point is at \((0,9)\) (x=0, y=9) and the second is at (9,1)? Wait, no, maybe the second yellow point is at (9,1)? Wait, no, when x=9, y=1? Wait, no, let's recalculate the slope. Wait, the line goes from (0,9) to (9,1). So the slope \(m=\frac{1 - 9}{9 - 0}=\frac{-8}{9}\)? Wait, no, that can't be. Wait, maybe I misread the points. Wait, the x - axis: the first yellow point is at x=0, y=9 (so (0,9)). The second yellow point: looking at the graph, when x=9, y=1? Wait, no, maybe the second point is (9,1)? Wait, but let's check the y - axis. Wait, the y - axis has 9 at the top, then 8,7,6,5,4,3,2,1,0. So (0,9) and (9,1). Then the slope is \(\frac{1 - 9}{9 - 0}=\frac{-8}{9}\)? Wait, no, that seems off. Wait, maybe the second point is (9,1)? Wait, no, maybe I made a mistake. Wait, let's use the slope - intercept form \(y=mx + b\). We know that when \(x = 0\), \(y = 9\), so \(b = 9\). Now, let's take another point. Let's say when \(x = 9\), \(y = 1\). So plug into \(y=mx + 9\): \(1=m(9)+9\). Then \(9m=1 - 9=-8\), so \(m=-\frac{8}{9}\)? Wait, that seems correct. Wait, but maybe the points are (0,9) and (9,1). So the equation is \(y=-\frac{8}{9}x + 9\)? Wait, no, wait, maybe I misread the second yellow point. Wait, the x - axis: the first yellow point is at x=0, y=9. The second yellow point: looking at the graph, at x=9, y=1? Wait, the y - coordinate at x=9 is 1? Let's check the grid. Each grid line is 1 unit. So from (0,9) to (9,1): the run is 9 (from x=0 to x=9), the rise is \(1 - 9=-8\). So slope is \(\frac{-8}{9}\). So the equation is \(y=-\frac{8}{9}x + 9\). Wait, but maybe the problem expects integer coefficients. Wait, maybe I made a mistake in the points. Wait, maybe the second yellow point is (9,0)? No, the yellow point at x=9 is at y=1? Wait, the graph shows a yellow point at (9,1) and (0,9). So let's proceed.

Wait, maybe the two yellow points are (0,9) and (9,1). So the slope \(m=\frac{1 - 9}{9 - 0}=\frac{-8}{9}\), and the y - intercept \(b = 9\) (since when \(x = 0\), \(y = 9\)). So the equation in slope - intercept form is \(y=-\frac{8}{9}x + 9\).

Wait, but maybe I misread the second point. Let's check again. The x - axis: 0,1,2,3,4,5,6,7,8,9,10. The y - axis: 9,8,7,6,5,4,3,2,1,0. The first yellow point is at (0,9) (x=0, y=9). The second yellow point is at (9,1) (x=9, y=1). So the slope is \(\frac{1 - 9}{9 - 0}=-\frac{8}{9}\), and the y - intercept is 9. So the equation is \(y = -\frac{8}{9}x+9\).

Wait, but maybe the problem has a typo or I misread the points. Alternatively, maybe the second point is (9,0)? Let's check. If the second point is (9,0), then slope is \(\frac{0 - 9}{9 - 0}=-1\), and equation is \(y=-x + 9\). But the yellow point at x=9: looking at the graph, the yellow point is at y=1, not 0. So maybe my initial reading is correct.

Wait, let's recalculate. \(x_1 = 0\), \(y_1 = 9\); \(x_2 = 9\), \(y_2 = 1\). Then \(m=\frac{1 - 9}{9 - 0}=\frac{-8}{9}\). So the equation is \(y=-\frac{8}{9}x + 9\).

Answer:

\(y = -\frac{8}{9}x + 9\)