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 font size 14 12 16 10 12 14 16 18 24 22 word count 352 461 340 407 435 381 280 201 138 114 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: Recall slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use the fact that when $x$ increases, $y$ (word count) decreases, so the slope $m$ is negative.

Step2: Use a point - slope approach (approximate)

Let's take two points, say $(10,407)$ and $(24,138)$. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{138 - 407}{24 - 10}=\frac{- 269}{14}\approx - 19.21$. But we can also estimate by looking at the general trend. As $x$ (font - size) changes from 10 to 24 (an increase of 14), $y$ (word - count) changes from 407 to 138 (a decrease of 269).
We can also use the fact that we can substitute points into the equations to check. Let's take the mid - range point, say $x = 16$. When $x = 16$, from the table $y\approx280$.
For $y=-55x + 407$, when $x = 16$, $y=-55\times16 + 407=-880 + 407=-473$ (way off).
For $y=-41x + 814$, when $x = 16$, $y=-41\times16+814=-656 + 814 = 158$ (off).
For $y=-38x + 922$, when $x = 16$, $y=-38\times16+922=-608 + 922 = 314$ (close).
For $y=-26x + 723$, when $x = 16$, $y=-26\times16+723=-416+723 = 307$ (also close).
If we consider the overall trend and more points, we know that the line should fit the data well. A more accurate way is to use the least - squares method for finding the line of best fit. But by estimating:
We know that as $x$ increases, $y$ decreases. The slope should be such that it reflects the rate of decrease.
Let's assume we use the fact that when $x$ is small (e.g., $x = 10$), $y\approx407$.
If we substitute $x = 10$ into $y=-38x + 922$, we get $y=-38\times10+922=-380 + 922 = 542$ (a bit high).
If we substitute $x = 10$ into $y=-26x + 723$, we get $y=-26\times10+723=-260 + 723 = 463$ (closer to 407 considering the overall trend).

Answer:

$y=-26x + 723$