Graphing Proportional Relationships: Practice

The constant of proportionality is the ratio $\frac{y}{x}$, taken at any pair with $x$ not zero.

$k = \frac{12}{4} = 3$

Multiplying gives $48$ and subtracting gives $8$; neither is the ratio.

Divide each $y$ by its $x$ and see whether the ratios match.

$\frac{5}{1} = \frac{10}{2} = \frac{15}{3} = 5$

That table has one ratio throughout. The table with $y = 5, 7, 9$ gives $5, 3.5, 3$, the one with $y = 1, 4, 9$ gives $1, 2, 3$, and the one with $y = 6, 10, 14$ gives $6, 5, \frac{14}{3}$. None of those match.

A proportional relationship is $y = kx$. Put in $x = 0$:

$y = k \times 0 = 0$

So the graph always contains the origin, whatever $k$ is. It passes through $(1, 1)$ only when $k = 1$.

Put $x = 5$ into the equation and multiply by the unit rate.

$y = 3 \times 5 = 15$

Five pounds cost $15$ dollars. Adding $3$ and $5$ gives $8$, which is not how a price per pound works.

Start from the ratio and multiply both sides by $x$, which is allowed because $x$ is not zero.

$\frac{y}{x} = 7 \quad\Longrightarrow\quad y = 7x$

The equation $x = 7y$ would make $\frac{y}{x}$ equal $\frac{1}{7}$, not $7$.

The units of $k$ are the units of $y$ per unit of $x$, because $k = \frac{y}{x}$.

$k = \frac{\text{liters}}{\text{hours}}$

So $k = 6$ liters per hour: the car uses $6$ liters in each hour of driving.

Put $x = 1$ into the equation.

$y = 4 \times 1 = 4$

The height at $x = 1$ is always $k$, which is why $k$ is the unit rate: the output for one unit of input.

A straight line through the origin, other than an axis, is the graph of $y = kx$. Find $k$ from the point that is not the origin.

$k = \frac{10}{2} = 5$

So the equation is $y = 5x$. Check: $5 \times 2 = 10$.

A point $(x, y)$ is on the graph when $y$ equals $3$ times $x$. Test $(4, 12)$:

$3 \times 4 = 12$

It works. For $(12, 4)$ the coordinates are swapped, $(4, 7)$ adds $3$ instead of multiplying, and $3 \times 3 = 9$, not $1$.

Finding $k$ means dividing $y$ by $x$. At the origin that division is

$\frac{0}{0},$

and division by zero has no answer. The origin is on every proportional graph; it just cannot tell you $k$. Use any other point.

The equation has the form $y = kx$ with $k = -3$, so its graph is a straight line through the origin. At $x = 1$:

$y = -3 \times 1 = -3$

Each step of $1$ across goes $3$ down, so the line falls from left to right.

The cost is $y = 5x$, which is proportional, so every point lies on the line through the origin. But $x$ must be a whole number of tickets, $0$ or more:

$(0, 0),\ (1, 5),\ (2, 10),\ \ldots$

So the graph is separate dots on that line, not the whole line.