Graphing Proportional Relationships: Additional Practice (Set 2)

The constant is

$k = \frac{6}{9} = \frac{2}{3}.$

Then $y = \frac{2}{3} \times 15 = 10$. Using $\frac{9}{6}$ gives $22.5$, the rate read backward, and adding $6$ to both coordinates gives $12$.

Solve $14x = 49$ by dividing both sides by $14$.

$x = 49 \div 14 = 3.5$

Check: $14 \times 3.5 = 49$. Multiplying $49$ by $14$ gives $686$, which answers no question here.

Test $(8, 6)$ by multiplying its first coordinate by $\frac{3}{4}$.

$\frac{3}{4} \times 8 = 6$

It works. The points $(6, 8)$ and $(3, 4)$ have ratio $\frac{4}{3}$, and $(12, 8)$ has ratio $\frac{2}{3}$.

Divide each $y$ by its $x$, using the sign rules for division.

$\frac{6}{-2} = \frac{-3}{1} = \frac{-12}{4} = -3$

The ratios match, so the table is proportional with $k = -3$. Falling values and negative inputs are both allowed.

The constant is $k = \frac{4}{10} = 0.4$, and at $x = 1$:

$y = 0.4 \times 1 = 0.4$

The height at $x = 1$ is the unit rate, the output for one unit of input. When $x = 0$ the amount is $0$, as on every proportional graph.

On the vertical axis every point has $x = 0$, while $y$ takes many values. But $y = kx$ gives

$y = k \times 0 = 0$

at $x = 0$, so it cannot produce those points. The other lines are $y = -5x$, $y = 0.5x$ and $y = x$.

Work out both costs for $5$ hours.

$\text{A: } 7 \times 5 = 35, \qquad \text{B: } 12 + 4 \times 5 = 32$

Plan B is cheaper. Plan A has the form $y = kx$, but Plan B costs $12$ dollars even for $0$ hours, so only Plan A is proportional.

The dive ends at $x = 6$ minutes.

$y = -12 \times 6 = -72$

So the graph ends at $(6, -72)$, in Quadrant IV: the submarine is $72$ meters below where it started. The point $(-6, 72)$ is on the full line, but negative times are not part of the dive.

Write the relationship as $y = kx$ and replace $x$ with $2x$.

$k \times (2x) = 2 \times (kx)$

The new output is $2$ times the old one. That is the pencils from the lesson: buy twice as many and you pay twice as much.

From the first pair, a proportional relationship would have $k = \frac{10}{4} = 2.5$. Then at $x = 6$:

$y = 2.5 \times 6 = 15$

The table says $14$, so the ratios $2.5$ and $\frac{14}{6}$ differ and it is not proportional.

Find each distance at $x = 40$, then subtract.

$7.5 \times 40 - 6 \times 40 = 300 - 240 = 60$

The gap grows by $1.5$ meters every second, so after $40$ seconds it is $60$ meters.

The constant is flour over eggs.

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

So $y = 2.5x$. Eggs come in whole numbers, $0$ or more, so the graph is dots on that line. The value $0.4$ is eggs per cup, the rate read the other way.