Proportionality
Texas 8th Grade Math · Learn by Concept
Help Questions
Texas 8th Grade Math › Proportionality
Points on a line: (2,7), (4,11), (6,15), (8,19)
Using any two points from the line, what is the slope (rate of change) $m$? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$2$
$\frac{1}{2}$
$-2$
$4$
Explanation
Pick two points, for example $(2,7)$ and $(6,15)$: $m=\frac{15-7}{6-2}=\frac{8}{4}=2$. Using another pair, $(4,11)$ and $(8,19)$: $m=\frac{19-11}{8-4}=\frac{8}{4}=2$. The slope is rise over run (change in $y$ over change in $x$), and it is constant for any two points on the same line. The right triangles you could draw between these point pairs have proportional legs (similar triangles), so the ratio $\frac{\text{rise}}{\text{run}}$ stays $2$.
Water flows at a constant rate. Points on the graph are $(1, 3.5)$, $(2, 7)$, $(3, 10.5)$, $(4, 14)$ where $x$ is minutes and $y$ is gallons.
What is the unit rate of flow?
7 gallons/minute
3.5 minutes/gallon
3.5 gallons/minute
0 gallons/minute
Explanation
Unit rate is the amount per 1 unit of $x$. The relationship is proportional ($y=kx$), so the slope equals the unit rate. Using any two points, slope $m=\frac{7-3.5}{2-1}=3.5$. Thus $y=3.5x$, and the unit rate is 3.5 gallons per minute.
Triangle ABC has vertices A(2,4), B(6,8), C(4,12). Triangle DEF has vertices D(1,2), E(3,4), F(2,6). What is the scale factor of the dilation from triangle ABC to triangle DEF?
2
$1/2$
$3/2$
3
Explanation
Similar figures have corresponding sides in proportion. Compare a corresponding coordinate pair: from A(2,4) to D(1,2), each coordinate is multiplied by $1/2$, so every side length is also multiplied by $1/2$. Checking another pair, B(6,8) to E(3,4) shows the same factor. The scale factor (new ÷ original) is $1/2$, and all corresponding ratios match.
A line is graphed on a coordinate plane. It crosses the y-axis at 4 and passes through the point (3, 10).
Which equation represents the graphed line?
$y = 4x + 2$
$y = 2x$
$y = 2x - 4$
$y = 2x + 4$
Explanation
The y-intercept is $b = 4$ (non-proportional since $b \ne 0$). Using the point (3, 10), the slope is $m = (10 - 4) / 3 = 2$. So the equation is $y = 2x + 4$. Options with $b = 0$ are proportional and do not match the intercept, and swapping $m$ and $b$ gives the wrong line.
A teacher compares study hours ($x$) to test scores ($y$). Approximate trendline: $y = 7.8x + 42$ for $2 \le x \le 10$.
Using the trendline, predict the test score when $x=6$, rounded to the nearest whole number.
47
89
90
136
Explanation
Substitute $x=6$ into $y=7.8x+42$: $y=7.8(6)+42=46.8+42=88.8\approx 89$. Slope-only mistake: $7.8\times 6=46.8\approx 47$ (ignores the intercept). Arithmetic slip: 90 (overestimates $7.8\times 6$ before adding 42). Extrapolation trap: 136 comes from $x=12$ (outside $2\le x\le 10$); predictions outside the data range are unreliable.
A cyclist rides at a constant speed. The relationship between time $x$ (hours) and distance $y$ (miles) is shown by the points $(1, 60)$, $(2, 120)$, and $(3, 180)$. Which equation represents this proportional relationship?
y = 180x
y = 60x + 10
x = 60y
y = 60x
Explanation
For a proportional relationship, $y = kx$ and $k = \frac{y}{x}$. Using any point: $k = \frac{60}{1} = 60$, $\frac{120}{2} = 60$, $\frac{180}{3} = 60$. The constant ratio $\frac{y}{x}$ is 60, so $y = 60x$. Adding a constant (like $+10$) would make it non-proportional because it would not pass through the origin.
Rectangle PQRS has dimensions 5 by 12 units. After dilation, rectangle P'Q'R'S' has dimensions 15 by 36 units. What is the scale factor of the dilation from rectangle PQRS to rectangle P'Q'R'S'?
$1/3$
$12/5$
$5/12$
3
Explanation
For similar figures, the scale factor equals new length ÷ original length for any corresponding sides. Using the widths: $15 \div 5 = 3$. Using the heights: $36 \div 12 = 3$. Since both corresponding ratios are equal, the scale factor is 3.
A line passes through (-3,1) and (5,9).
What is the slope between these two points? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$8$
$1$
$-1$
$2$
Explanation
Compute rise over run: $m=\frac{9-1}{5-(-3)}=\frac{8}{8}=1$. On this line, moving $8$ units right increases $y$ by $8$ units, so the rate of change is $1$ unit up per $1$ unit right. Any other segment along the same line forms a right triangle similar to this one, so $\frac{\text{rise}}{\text{run}}$ remains $1$.
A car travels at a constant speed modeled by $y = 65x$, where $x$ is hours and $y$ is miles.
How does the unit rate appear on the graph of this relationship?
The slope of the line is 65, and the line passes through the origin.
The $y$-intercept is 65, and the slope is 0.
The slope is $\tfrac{1}{65}$ and the line crosses the $y$-axis at 65.
The graph is a curve that increases as $x$ increases.
Explanation
For proportional relationships $y=kx$, the graph is a line through the origin with slope $k$. Here $k=65$, so the unit rate is 65 miles per hour and equals the slope of the line.
The cost of gasoline is proportional to the gallons purchased. If 5 gallons cost 17.50 and 8 gallons cost 28.00, which equation represents the relationship where $y$ is the total cost (dollars) and $x$ is the number of gallons?
y = 3.5x
y = 3.5x + 1
x = 3.5y
y = 0.35x
Explanation
In a proportional relationship, $y = kx$. Compute $k$ using $k = \frac{y}{x}$: $k = \frac{17.50}{5} = 3.5$ and $\frac{28.00}{8} = 3.5$. Since the ratio $\frac{y}{x}$ is constant, the equation is $y = 3.5x$. Adding a constant (like $+1$) would make it non-proportional.