Question

u.s. population
a $3x + 150$
b $18x + 235$
c $5x + 236$
d $13x + 270$

Explanation:

Step1: Estimate the slope

Take two points, say when \(x = 1\), \(y\approx240\); when \(x = 6\), \(y\approx250\).
Slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{250 - 240}{6 - 1}=\frac{10}{5} = 2\) (approximate visual check). Let's check with another method. Assume a linear form \(y=mx + b\).
If \(x = 1\), for option B: \(y=18\times1+235=253\); for option C: \(y = 5\times1+236=241\); for option A: \(y=3\times1+150 = 153\); for option D: \(y=13\times1+270=283\).
When \(x = 6\), option B: \(y=18\times6+235=108 + 235=343\) (too high). Option C: \(y=5\times6+236=30 + 236=266\). Option A: \(y=3\times6+150=18 + 150=168\) (too low). Option D: \(y=13\times6+270=78+270 = 348\) (too high).
Another way: when \(x = 1\), the \(y -\)value (population) is around \(240\). Plug \(x = 1\) into \(y=mx + b\).
For \(y = 18x+235\), when \(x = 1\), \(y=18 + 235=253\); for \(y=5x + 236\), when \(x=1\), \(y=5+236 = 241\); for \(y=3x+150\), \(y=153\); for \(y=13x + 270\), \(y=283\).
When \(x=6\), \(y = 5x+236=5\times6+236=266\). If we assume \(x = 1\) (year \(1\)) population \(y_1\approx240\), \(x = 6\) (year \(6\)) population \(y_2\approx265\).
Slope \(m=\frac{265 - 240}{6 - 1}=\frac{25}{5}=5\). The \(y -\)intercept \(b\): using \(y=mx + b\) and \(x = 1\), \(y=241\) (\(m = 5\)), \(241=5\times1+b\), \(b=236\).

Answer:

C. \(5x + 236\)