Proportional Graphs - Pre Algebra

1. A proportional relationship's graph passes through $(3, 12)$. What is the value of $y$ when $x = 1$?

   $4$. In a proportional relationship $\frac{y}{x}$ is constant, so $\frac{12}{3} = 4$ and the point $(1, 4)$ is on the line.

2. A proportional relationship's graph passes through $(2, 7)$. What is the $y$-value on the graph when $x = 6$?

   $21$. Tripling $x$ from 2 to 6 triples $y$: $7 \times 3 = 21$.

3. On the graph of a proportional relationship, what is true about $\frac{y}{x}$ for every point other than the origin?

   It is the same value for every point. That shared ratio is the constant of proportionality $k$.

4. What two features must a graph have to show a proportional relationship?

   It must be a straight line, and it must pass through the origin. A line that misses the origin or a curve is not proportional.

5. Two proportional relationships are graphed on the same axes. What does the steeper line show?

   A greater constant of proportionality, meaning a greater unit rate ($y$ increases more for each 1 unit of $x$).

6. Which point must the graph of every proportional relationship pass through?

   The origin, $(0, 0)$. In $y = kx$, an input of 0 always gives an output of 0.

7. Which point lies on the same proportional graph as $(5, 15)$: $(2, 6)$ or $(2, 7)$?

   $(2, 6)$. Its ratio $\frac{6}{2} = 3$ matches $\frac{15}{5} = 3$, while $\frac{7}{2} = 3.5$ does not.

8. In a proportional cost graph with pounds on the $x$-axis and cost on the $y$-axis, what does the point $(0, 0)$ mean?

   Buying 0 pounds costs 0 dollars. There is no starting fee, which is why a proportional graph begins at the origin.

9. A proportional graph of distance in miles versus time in hours passes through $(2, 90)$. What unit rate does the graph show?

   45 miles per hour. The point means 90 miles in 2 hours, and $\frac{90}{2} = 45$.

10. A straight line passes through $(0, 2)$ and $(4, 10)$. Why does it not show a proportional relationship?

    It crosses the $y$-axis at 2, not at the origin. A proportional graph must contain $(0, 0)$.

11. A proportional graph passes through $(4, 6)$. What is the $x$-value on the graph when $y = 3$?

    $2$. Halving $y$ from 6 to 3 halves $x$ from 4 to 2.

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