Question

rob is investigating the effects of font size on the number of words that fit on a page. he changes the font size on an essay and records the number of words on one page of the essay. the table shows his data. words per page which equation represents the approximate line of best fit for data, where x represents font size and y represents the number of words on one page? $y=-55x + 407$ $y=-41x + 814$ $y=-38x + 922$ $y=-26x + 723$

Explanation:

Step1: Calculate the slope and y - intercept approximately

We can use two points \((x_1,y_1)\) and \((x_2,y_2)\) to estimate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1 = 10,y_1=407)\) and \((x_2 = 24,y_2 = 138)\)

$$m=\frac{138 - 407}{24 - 10}=\frac{-269}{14}\approx - 19.21$$

Another way is to check the trend. As \(x\) (font size) increases, \(y\) (word count) decreases.
We can also use the formula for the line of best - fit \(y=mx + b\).
Let's check each option by substituting a value of \(x\).
For option \(y=-41x + 814\):
When \(x = 10\), \(y=-41\times10+814=-410 + 814=404\) (close to \(407\))
When \(x = 12\), \(y=-41\times12 + 814=-492+814 = 322\) (not very close to \(461\) and \(435\))
For option \(y=-38x + 922\):
When \(x = 10\), \(y=-38\times10+922=-380 + 922 = 542\) (not close to \(407\))
For option \(y=-26x + 723\):
When \(x = 10\), \(y=-26\times10+723=-260+723 = 463\) (close to \(407\) but let's check another value)
When \(x = 14\), \(y=-26\times14+723=-364 + 723=359\) (close to \(352\) and \(381\))
When \(x = 16\), \(y=-26\times16+723=-416+723 = 307\) (not close to \(340\) and \(280\))
For option \(y=-41x + 814\):
When \(x = 16\), \(y=-41\times16+814=-656+814 = 158\) (not close)
Let's use a better approach.
We know that the general form of a line is \(y=mx + b\).
We can calculate the mean of \(x\) values \(\bar{x}=\frac{14 + 12+16+10+12+14+16+18+24+22}{10}=\frac{178}{10}=17.8\)
The mean of \(y\) values \(\bar{y}=\frac{352+461+340+407+435+381+280+201+138+114}{10}=\frac{3009}{10}=300.9\)
We can also use the formula \(m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\) (a more complex calculation, but for approximation)
Another way:
Let's check the option \(y=-26x + 723\)
When \(x = 10\), \(y=-26\times10+723 = 463\)
When \(x=12\), \(y=-26\times12 + 723=-312+723 = 411\)
When \(x = 14\), \(y=-26\times14+723=-364+723=359\)
When \(x = 16\), \(y=-26\times16+723=-416+723 = 307\)
When \(x = 18\), \(y=-26\times18+723=-468+723 = 255\)
When \(x = 22\), \(y=-26\times22+723=-572+723 = 151\)
When \(x = 24\), \(y=-26\times24+723=-624+723=99\)

Let's check option \(y=-41x + 814\)
When \(x = 10\), \(y=-41\times10 + 814=404\)
When \(x = 12\), \(y=-41\times12+814=-492 + 814=322\)
When \(x = 14\), \(y=-41\times14+814=-574+814 = 240\) (not good)

Let's check option \(y=-38x + 922\)
When \(x = 10\), \(y=-38\times10+922=542\) (not good)

Let's check option \(y=-55x + 407\)
When \(x = 10\), \(y=-55\times10+407=-143\) (not possible)

Answer:

\(y=-26x + 723\)