A proportional relationship has the form $y = kx$ (so $b=0$) and passes through the origin. Function 1 is $y=3x$ with $b=0$, so it is proportional. Function 2 is $y=2x+7$ with $b=7\neq0$, so it is non-proportional.

A dilation with scale factor k affects linear measurements by ×k and area by ×k^2. Here k = 1.5, so perimeter scales by 1.5 and area scales by 1.5^2 = 2.25. Angle measures, shape, and parallel sides are preserved.

For a dilation centered at the origin with scale factor k, use the rule $(x,y) \to (kx, ky)$. Here $k=2.5$. Compute each vertex: (2,3) → (2.5·2, 2.5·3) = (5, 7.5); (6,1) → (15, 2.5); (4,7) → (10, 17.5). Because each coordinate is multiplied by the same positive factor, side lengths scale by 2.5, angles stay the same, and the figure is an enlargement centered at the origin.

Pick two points, for example $(2,7)$ and $(6,15)$: $m=\frac{15-7}{6-2}=\frac{8}{4}=2$. Using another pair, $(4,11)$ and $(8,19)$: $m=\frac{19-11}{8-4}=\frac{8}{4}=2$. The slope is rise over run (change in $y$ over change in $x$), and it is constant for any two points on the same line. The right triangles you could draw between these point pairs have proportional legs (similar triangles), so the ratio $\frac{\text{rise}}{\text{run}}$ stays $2$.

Unit rate is the amount per 1 unit of $x$. The relationship is proportional ($y=kx$), so the slope equals the unit rate. Using any two points, slope $m=\frac{7-3.5}{2-1}=3.5$. Thus $y=3.5x$, and the unit rate is 3.5 gallons per minute.

Compute slope with any two points, say $(1,2)$ and $(3,8)$: $m=\frac{8-2}{3-1}=\frac{6}{2}=3$. Use $b=y-mx$ with $(1,2)$: $b=2-3(1)=-1$. So $(m,b)=(3,-1)$.

For a proportional relationship, $y = kx$ and $k = \frac{y}{x}$. Using any point: $k = \frac{60}{1} = 60$, $\frac{120}{2} = 60$, $\frac{180}{3} = 60$. The constant ratio $\frac{y}{x}$ is 60, so $y = 60x$. Adding a constant (like $+10$) would make it non-proportional because it would not pass through the origin.

The y-intercept is $b = 4$ (non-proportional since $b \ne 0$). Using the point (3, 10), the slope is $m = (10 - 4) / 3 = 2$. So the equation is $y = 2x + 4$. Options with $b = 0$ are proportional and do not match the intercept, and swapping $m$ and $b$ gives the wrong line.

Substitute $x=6$ into $y=7.8x+42$: $y=7.8(6)+42=46.8+42=88.8\approx 89$. Slope-only mistake: $7.8\times 6=46.8\approx 47$ (ignores the intercept). Arithmetic slip: 90 (overestimates $7.8\times 6$ before adding 42). Extrapolation trap: 136 comes from $x=12$ (outside $2\le x\le 10$); predictions outside the data range are unreliable.

For direct variation, $C=kn$. Find $k=\frac{C}{n}=\frac{18}{6}=3$, so $C=3n$. The other options invert the relationship, add a nonzero intercept, or solve for the wrong variable.