question\nthe table shows the total number of posts made by the average registered user of a new social…

Explanation:

Step1: Recall linear - equation form

The general form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. In our case, $y=p(x)$ and $x$ is the day.

Step2: Calculate the slope $m$

The formula for the slope $m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$. First, find the means of $x$ and $y$ values. The $x$ - values are $x={0,1,2,3,4,5,6,7}$, so $\bar{x}=\frac{0 + 1+2+3+4+5+6+7}{8}=\frac{28}{8}=3.5$. The $y$ - values are $y = {0,3,5,8,10,12,15,17}$, so $\bar{y}=\frac{0 + 3+5+8+10+12+15+17}{8}=\frac{70}{8}=8.75$. $\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=(0 - 3.5)(0 - 8.75)+(1 - 3.5)(3 - 8.75)+(2 - 3.5)(5 - 8.75)+(3 - 3.5)(8 - 8.75)+(4 - 3.5)(10 - 8.75)+(5 - 3.5)(12 - 8.75)+(6 - 3.5)(15 - 8.75)+(7 - 3.5)(17 - 8.75)$ $=(-3.5)(-8.75)+(-2.5)(-5.75)+(-1.5)(-3.75)+(-0.5)(-0.75)+(0.5)(1.25)+(1.5)(3.25)+(2.5)(6.25)+(3.5)(8.25)$ $=30.625 + 14.375+5.625 + 0.375+0.625+4.875+15.625+28.875$ $=101$. $\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=(0 - 3.5)^{2}+(1 - 3.5)^{2}+(2 - 3.5)^{2}+(3 - 3.5)^{2}+(4 - 3.5)^{2}+(5 - 3.5)^{2}+(6 - 3.5)^{2}+(7 - 3.5)^{2}$ $=12.25+6.25 + 2.25+0.25+0.25+2.25+6.25+12.25$ $=42$. $m=\frac{101}{42}\approx2.4$.

Step3: Calculate the y - intercept $b$

We know that $y = mx + b$. Substitute one of the points, say $(0,0)$ (we can also use other points) into the equation $y=mx + b$. When $x = 0$ and $y = 0$, and $m\approx2.4$, we get $0=2.4\times0 + b$, so $b\approx - 0.5$.

Answer:

$p(x)=2.4x - 0.5$