Graphing Proportional Relationships: Practice

Find the constant first.

$k = \frac{10}{4} = 2.5$

Then $y = 2.5 \times 14 = 35$. Adding $10$ to both coordinates would give $20$, which keeps the difference but not the ratio.

Compute $\frac{y}{x}$ for each point.

$\frac{3}{2} = \frac{9}{6} = \frac{15}{10} = 1.5$

Those three lie on $y = 1.5x$. For $(4, 7)$ the ratio is $\frac{7}{4} = 1.75$, so it is off that line.

The constant is

$k = \frac{4}{6} = \frac{2}{3}.$

Then $y = \frac{2}{3} \times 15 = 10$. Keeping $k$ as a fraction avoids a rounded decimal. Using $\frac{6}{4}$ instead gives $22.5$, the rate read backward.

Solve $-0.5x = -12$ by dividing both sides by $-0.5$.

$x = \frac{-12}{-0.5} = 24$

A negative divided by a negative is positive, so the answer is $24$ minutes. Check: $-0.5 \times 24 = -12$.

Read such a line's height at $x = 1$ and call it $k$. Every $1$ across changes the height by that same amount, so

$y = kx.$

The axes are excluded: on the vertical axis $x$ is always $0$, and on the horizontal one $k$ would be $0$. The line may fall, and $k$ may be any nonzero number.

Each step adds $3$ to $x$ and $9$ to $y$, so the steps are equal. But the ratios are

$\frac{9}{2} = 4.5, \quad \frac{18}{5} = 3.6, \quad \frac{27}{8} = 3.375,$

which do not match. Running the pattern back to $x = 0$ gives $y = 3$, not $0$.

Steepness is the unit rate, so compare miles per hour.

$\frac{10}{4} = 2.5, \qquad \frac{9}{3} = 3$

B walks faster, so B's line climbs more for each hour across. The totals alone do not decide it, because the times differ.

Steepness is the size of $k$, its absolute value, and the sign gives the direction.

$|-3| = 3 > 2 = |2|$

So $y = -3x$ is steeper, and because its $k$ is negative it falls from left to right.

Divide $y$ by $x$.

$k = \frac{7.5}{2.5} = 3$

So $y = 3x$. The equation $y = x + 5$ also fits this one point, but it is not proportional: at $x = 0$ it gives $5$.

Both prices give the same unit rate.

$\frac{12}{3} = \frac{20}{5} = 4$

So $y = 4x$, and $8$ notebooks cost $4 \times 8 = 32$ dollars.

Compare the ratios at the points other than the origin.

$\frac{2}{1} = 2, \quad \frac{8}{2} = 4, \quad \frac{18}{3} = 6$

They differ, so it is not proportional. The graph does start at the origin, but its points bend upward instead of lying on a straight line.

The distance is $y = 80x$. Time runs from $0$ to $2.5$ hours, every value in between included, and the trip ends at

$80 \times 2.5 = 200.$

So the graph is the solid piece of the line from $(0, 0)$ to $(2.5, 200)$. Negative times and times past the trip are not part of it.