Question

a person 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 two - point method to estimate the slope or check the trend of the data. As the font size ($x$) increases, the word count ($y$) decreases, so the slope $m$ should be negative.

Step2: Select two points

Let's take two points from the data, 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 check by substituting $x$ values into the given equations and seeing which one gives values closest to the actual $y$ values in the data.
Let's take $x = 14$.
For $y=-55x + 407$, when $x = 14$, $y=-55\times14 + 407=-770+407=-363$ (way off).
For $y=-41x + 814$, when $x = 14$, $y=-41\times14 + 814=-574 + 814 = 240$ (off).
For $y=-38x + 922$, when $x = 14$, $y=-38\times14+922=-532 + 922=390$.
For $y=-26x + 723$, when $x = 14$, $y=-26\times14 + 723=-364+723 = 359$ which is close to the actual $y$ - value of 352 when $x = 14$.
We can check more points. For example, when $x = 12$:
For $y=-38x + 922$, $y=-38\times12+922=-456 + 922 = 466$ (close to 461).
For $y=-26x + 723$, $y=-26\times12+723=-312 + 723=411$ (not as close as $y=-38x + 922$ for this point).

Answer:

$y=-38x + 922$