Question

16.
input values
8
4
0
-4
-8
output values
2
8
14
20
26
constant rate of change (slope):
equation:

Explanation:

Step1: Find the slope (rate of change)

We can take two points from the input (x) and output (y) values. Let's take (8, 2) and (4, 8). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
So, $m=\frac{8 - 2}{4 - 8}=\frac{6}{-4}=-\frac{3}{2}$? Wait, no, maybe I mixed up input and output. Wait, input is x, output is y. Let's list the pairs: (8,2), (4,8), (0,14), (-4,20), (-8,26). Let's take (8,2) and (4,8). So $x_1 = 8, y_1 = 2$; $x_2 = 4, y_2 = 8$. Then slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{8 - 2}{4 - 8}=\frac{6}{-4}=-\frac{3}{2}$? Wait, but let's check another pair: (4,8) and (0,14). $x_1 = 4, y_1 = 8$; $x_2 = 0, y_2 = 14$. Then $m=\frac{14 - 8}{0 - 4}=\frac{6}{-4}=-\frac{3}{2}$. Wait, but when x decreases by 4 (from 8 to 4), y increases by 6 (from 2 to 8). Wait, maybe I had x and y reversed. Wait, input is x, output is y. So when x=8, y=2; x=4, y=8; x=0, y=14; x=-4, y=20; x=-8, y=26. So let's take x=8, y=2 and x=0, y=14. Then slope $m=\frac{14 - 2}{0 - 8}=\frac{12}{-8}=-\frac{3}{2}$? No, that can't be. Wait, wait, when x decreases by 8 (from 8 to 0), y increases by 12 (from 2 to 14). So 12/-8 = -3/2. But let's check x=0, y=14 and x=-4, y=20. x decreases by 4, y increases by 6. 6/-4 = -3/2. So slope is -3/2? Wait, but let's use the slope formula correctly. Let's take two points: (x1, y1) = (8, 2) and (x2, y2) = (4, 8). Then $m = (y2 - y1)/(x2 - x1) = (8 - 2)/(4 - 8) = 6/(-4) = -3/2$. Wait, but when x increases by -4 (decreases by 4), y increases by 6. So the rate of change is -3/2? Wait, but let's check the equation. Let's use point-slope form. Let's take the point (0,14), which is the y-intercept (since x=0, y=14). So the equation of a line is $y = mx + b$, where b is the y-intercept. We found m = -3/2? Wait, no, wait, when x=0, y=14, so b=14. Let's check with x=8: $y = -\frac{3}{2}(8) + 14 = -12 + 14 = 2$, which matches. x=4: $y = -\frac{3}{2}(4) + 14 = -6 + 14 = 8$, which matches. x=-4: $y = -\frac{3}{2}(-4) + 14 = 6 + 14 = 20$, which matches. x=-8: $y = -\frac{3}{2}(-8) + 14 = 12 + 14 = 26$, which matches. So the slope (constant rate of change) is $-\frac{3}{2}$? Wait, but the problem says "constant rate of change (slope)". Wait, maybe I made a mistake in the sign. Wait, when x increases by 4 (from -8 to -4), y increases by 6 (from 26 to 20? No, wait x=-8, y=26; x=-4, y=20. Oh! Wait, I had the pairs reversed. Input is x, output is y. So (x,y) pairs: (8,2), (4,8), (0,14), (-4,20), (-8,26). Wait, when x decreases (from 8 to 4 to 0 to -4 to -8), y increases (from 2 to 8 to 14 to 20 to 26). So as x increases (goes from -8 to -4 to 0 to 4 to 8), y decreases (from 26 to 20 to 14 to 8 to 2). So the slope is (y2 - y1)/(x2 - x1) when x2 > x1. Let's take x1=-8, y1=26; x2=-4, y2=20. Then $m=(20 - 26)/(-4 - (-8))=(-6)/(4)= -3/2$. Yes, that's correct. So the slope is -3/2. Wait, but let's check with x1=-4, y1=20; x2=0, y2=14. $m=(14 - 20)/(0 - (-4))=(-6)/4= -3/2$. Correct. So the constant rate of change (slope) is -3/2? Wait, but the problem says "constant rate of change (slope)". Wait, maybe I messed up input and output. Wait, input values: 8,4,0,-4,-8; output values:2,8,14,20,26. So when input (x) decreases by 4 (8→4), output (y) increases by 6 (2→8). So the rate of change is (change in y)/(change in x) = 6/(-4) = -3/2. Yes. So slope is -3/2.

Step2: Find the equation of the line

We know the slope $m = -\frac{3}{2}$ and the y-intercept (when x=0, y=14), so the equation is $y = mx + b$, where $b = 14$. So $y = -\frac{3}{2}x + 14$. Let's verify with x=8: $y = -\frac{3}{2}(8) + 14 = -12 + 14 = 2$, correct. x=4: $y = -\frac{3}{2}(4) + 14 = -6 + 14 = 8$, correct. x=-8: $y = -\frac{3}{2}(-8) + 14 = 12 + 14 = 26$, correct.

Wait, but wait, maybe I made a mistake in the slope sign. Let's check again. If x increases by 4 (from 4 to 8), y decreases by 6 (from 8 to 2). So (2 - 8)/(8 - 4) = (-6)/4 = -3/2. Yes, that's correct. So the slope is -3/2, and the equation is $y = -\frac{3}{2}x + 14$.

Answer:

Constant rate of change (slope): $-\frac{3}{2}$
Equation: $y = -\frac{3}{2}x + 14$