Graphing Proportional Relationships: Additional Practice (Set 1)

The constant is

$k = \frac{21}{6} = 3.5.$

Then $y = 3.5 \times 10 = 35$. Adding $4$ to both coordinates gives $25$, which does not keep the ratio.

The constant is $k = \frac{6}{8} = 0.75$. Test $(4, 3)$:

$0.75 \times 4 = 3$

It is half of the trip to $(8, 6)$. The point $(6, 8)$ swaps the coordinates, $(9, 7)$ adds $1$ to each, and $(8, 0)$ is on the x-axis, which only the origin shares with the graph.

The unit rate is $k = \frac{98}{7} = 14$ liters per minute, so $y = 14x$. Solve $14x = 252$ by dividing by $14$.

$x = 252 \div 14 = 18$

Multiplying $252$ by $14$ answers a different question.

In the table with $y = 0, 2.4, 5.6$, the input $0$ gives $0$ and the other ratios match:

$\frac{2.4}{3} = \frac{5.6}{7} = 0.8$

The table starting at $y = 1$ fails at $x = 0$. The one with $y = 2, 4, 8$ has ratios $2, \frac{4}{3}, \frac{8}{7}$, and the one with $y = 1, 4, 9$ has ratios $\frac{1}{3}, \frac{2}{3}, 1$.

Divide $y$ by $x$: $k = \frac{2}{5} = 0.4$. Then

$y = 0.4 \times 12 = 4.8.$

The value $2.5$ is $\frac{x}{y}$, the rate read backward. Adding $7$ to $2$ gives $9$, which ignores the ratio.

Put $x = 1$ into the equation.

$y = -1.5 \times 1 = -1.5$

In each second the height changes by $-1.5$ meters, a drop of $1.5$ meters. The sign gives the direction and the size gives how fast. Nothing here says where the balloon started.

The cost is $3$ dollars per class on top of the $10$-dollar fee. With no classes:

$\text{cost} = 10 + 3 \times 0 = 10$

The points rise by a steady $3$, so they lie on a straight line, but it meets the y-axis at $(0, 10)$, not the origin. Classes come in whole numbers, so the graph is dots.

Steepness is the size of $k$, ignoring its sign.

$|0.5| < |3| < |-4|$

So the order is $y = 0.5x$, $y = 3x$, $y = -4x$. The negative $k$ only makes its line fall; it does not make it less steep.

A proportional table needs equal ratios, not equal steps.

$\frac{6}{1} = 6, \quad \frac{10}{2} = 5, \quad \frac{14}{3} \approx 4.67$

The ratios differ. Running the steps back to $x = 0$ gives $y = 2$, a starting amount, which is why the table fails.

The constant is

$k = \frac{5}{-2} = -2.5.$

Then $y = -2.5 \times 4 = -10$. The line falls from left to right, so a positive $x$ gives a negative $y$.

Gasoline at a fixed price gives $\text{cost} = (\text{price per gallon}) \times (\text{gallons})$, and $0$ gallons cost $0$:

$y = kx$

The taxi fare and the plant both have a starting amount, so their graphs miss the origin. A square's area starts at the origin but grows along a curve.

The cost is $y = 0.15x$, and the largest input is $x = 40$.

$y = 0.15 \times 40 = 6$

So the graph ends at $(40, 6)$: pages across, dollars up. The point $(6, 40)$ swaps the coordinates.