Proportionality>Graphing Proportional Relationships and Interpreting Unit Rate as Slope(TEKS.Math.8.4.B)
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Texas 8th Grade Math › Proportionality>Graphing Proportional Relationships and Interpreting Unit Rate as Slope(TEKS.Math.8.4.B)
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.
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.
A recipe uses 2.5 cups of sugar per batch of muffins. Sample points are $(1, 2.5)$, $(3, 7.5)$, $(5, 12.5)$ where $x$ is batches and $y$ is cups of sugar.
What is the constant of proportionality (unit rate)?
0 cups per batch
0.4 cups per batch
5 cups per batch
2.5 cups per batch
Explanation
In a proportional relationship $y=kx$, the constant of proportionality $k=\frac{y}{x}$. Using any point: $\frac{2.5}{1}=2.5$ and $\frac{7.5}{3}=2.5$. The slope (unit rate) is 2.5 cups per batch, and the graph passes through the origin.
A line modeling a proportional relationship passes through the origin and the points $(2, 10)$ and $(5, 25)$.
What is the unit rate of this relationship?
2
5
10
0
Explanation
For a proportional line through the origin, the unit rate equals the slope: $m=\frac{25-10}{5-2}=\frac{15}{3}=5$. Thus $y=5x$, and the unit rate is 5 units of $y$ per 1 unit of $x$.
A taxi charges according to $y = 3 + 2x$, where $x$ is miles and $y$ is dollars.
Which statement is true about the unit rate and the graph?
The unit rate is 3, and the line passes through the origin.
The unit rate is 2, and the line does not pass through the origin.
The unit rate is 5, and the line is horizontal.
The unit rate is 2, and the line passes through the origin.
Explanation
The slope is 2 dollars per mile, which is the rate of change. The $+3$ is a starting fee (the $y$-intercept), so the graph does not pass through the origin and is not proportional. In proportional relationships $y=kx$, there is no constant term.