Question

find the common differences among the x-values and y-values. then, answ

1. {(3, 0), (6, 3), (9, 6), (12, 9)}

x-values
y-values
do they represent a linear function?

1. {(-2, -5), (-5, -11), (-8, -17), (-11, -20)}

x-values
y-values
do they represent a linear function?

Explanation:

Problem 5

Step1: Find x - differences

The x - values are 3, 6, 9, 12. Calculate the differences: $6 - 3 = 3$, $9 - 6 = 3$, $12 - 9 = 3$.

Step2: Find y - differences

The y - values are 0, 3, 6, 9. Calculate the differences: $3 - 0 = 3$, $6 - 3 = 3$, $9 - 6 = 3$.

Step3: Determine linear function

Since the common difference of x - values is constant (3) and the common difference of y - values is constant (3), the points represent a linear function.

Step1: Find x - differences

The x - values are - 2, - 5, - 8, - 11. Calculate the differences: $-5-(-2)=-3$, $-8 - (-5)=-3$, $-11-(-8)=-3$.

Step2: Find y - differences

The y - values are - 5, - 11, - 17, - 20. Calculate the differences: $-11-(-5)=-6$, $-17-(-11)=-6$, $-20 - (-17)=-3$. Wait, no, let's recalculate: $-11-(-5)=-6$, $-17 - (-11)=-6$, $-20-(-17)=-3$? Wait, no, $-20-(-17)=-3$ is wrong. Wait, $-17$ to $-20$: $-20-(-17)=-3$? Wait, no, the y - values are - 5, - 11, - 17, - 20. So $-11-(-5)=-6$, $-17 - (-11)=-6$, $-20-(-17)=-3$? Wait, that's a mistake. Wait, $-5$ to $-11$: difference is $-11 + 5=-6$; $-11$ to $-17$: $-17 + 11=-6$; $-17$ to $-20$: $-20 + 17=-3$. Wait, but for a linear function, the y - differences should be constant. Wait, no, maybe I made a mistake. Wait, the y - values: - 5, - 11, - 17, - 20. Let's check the slope between consecutive points. The slope between $(-2,-5)$ and $(-5,-11)$ is $\frac{-11 - (-5)}{-5-(-2)}=\frac{-6}{-3}=2$. The slope between $(-5,-11)$ and $(-8,-17)$ is $\frac{-17-(-11)}{-8 - (-5)}=\frac{-6}{-3}=2$. The slope between $(-8,-17)$ and $(-11,-20)$ is $\frac{-20-(-17)}{-11 - (-8)}=\frac{-3}{-3}=1$. Wait, that's a problem. Wait, no, my calculation of y - differences was wrong. Wait, $-17$ to $-20$: $-20-(-17)=-3$, but the previous differences were - 6. Wait, no, the points are $(-2,-5)$, $(-5,-11)$, $(-8,-17)$, $(-11,-20)$. Let's recalculate y - differences:

First difference: $-11-(-5)=-6$

Second difference: $-17 - (-11)=-6$

Third difference: $-20-(-17)=-3$

Wait, that means the y - differences are not constant. But wait, the slope between $(-2,-5)$ and $(-5,-11)$ is $\frac{-11 + 5}{-5 + 2}=\frac{-6}{-3}=2$

Slope between $(-5,-11)$ and $(-8,-17)$ is $\frac{-17 + 11}{-8 + 5}=\frac{-6}{-3}=2$

Slope between $(-8,-17)$ and $(-11,-20)$ is $\frac{-20 + 17}{-11 + 8}=\frac{-3}{-3}=1$

Wait, this is a contradiction. Wait, maybe I misread the y - value of the last point. The last point is $(-11,-20)$? Let's check the original problem: $\{(-2, - 5), (-5, - 11), (-8, - 17), (-11, - 20)\}$. So y - values are - 5, - 11, - 17, - 20.

Wait, $-20-(-17)=-3$, but the first two differences are - 6. So the y - differences are not constant. But wait, the x - differences are constant (- 3). For a function to be linear, the rate of change (slope) should be constant. The slope between the first two points is $\frac{-11 - (-5)}{-5-(-2)}=\frac{-6}{-3}=2$. The slope between the second and third points is $\frac{-17 - (-11)}{-8 - (-5)}=\frac{-6}{-3}=2$. The slope between the third and fourth points is $\frac{-20 - (-17)}{-11 - (-8)}=\frac{-3}{-3}=1$. Since the slope is not constant, the points do not represent a linear function. Wait, but my calculation of the third difference was wrong? Wait, $-17$ to $-20$: $-20-(-17)=-3$, but $-11$ to $-17$ is $-6$, $-5$ to $-11$ is $-6$. So the y - differences are - 6, - 6, - 3. Which are not constant. So the x - differences are constant (- 3), y - differences are not constant. So the function is not linear.

Wait, maybe I made a mistake in the y - value of the last point. Let me check again. The last point is $(-11, - 20)$? If it was $(-11, - 23)$, then the difference would be - 6. Maybe a typo? But according to the given problem, it's $(-11, - 20)$. So proceeding with the given values.

Step1: Find x - differences

x - values: - 2, - 5, - 8, - 11. Differences: $-5-(-2)=-3$, $-8 - (-5)=-3$, $-11-(-8)=-3$.

Step2: Find y - differences

y - values: - 5, - 11, - 17, - 20. Differences: $-11-(-5)=-6$, $-17 - (-11)=-6$, $-20-(-17)=-3$.

Step3: Determine linear function

Si…

Answer:

x - values differences: 3, 3, 3
y - values differences: 3, 3, 3
Do they represent a linear function? Yes

Problem 6