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.

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.

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 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.

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.

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.

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 $(m,b)=(3,-1)$.

The rate of change (slope) is the monthly fee, 30 dollars per month, so $m=30$. The starting value (y-intercept) is the one-time joining fee, 50 dollars, so $b=50$. Therefore $y=30x+50$. This connects to the context: $30x$ is the total monthly charges and $50$ is the initial fee.

Similar figures have corresponding sides in proportion. Compare a corresponding coordinate pair: from A(2,4) to D(1,2), each coordinate is multiplied by $1/2$, so every side length is also multiplied by $1/2$. Checking another pair, B(6,8) to E(3,4) shows the same factor. The scale factor (new ÷ original) is $1/2$, and all corresponding ratios match.

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.