Proportionality>Distinguishing Between Proportional and Non-Proportional Situations(TEKS.Math.8.5.F)
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Texas 8th Grade Math › Proportionality>Distinguishing Between Proportional and Non-Proportional Situations(TEKS.Math.8.5.F)
Function 1: $y = 3x$. Function 2: $y = 2x + 7$. Which relationship is proportional?
Only Function 1
Only Function 2
Both
Neither
Explanation
A proportional relationship has the form $y = kx$ (so $b=0$) and passes through the origin. Function 1 is $y=3x$ with $b=0$, so it is proportional. Function 2 is $y=2x+7$ with $b=7\neq0$, so it is non-proportional.
Scenario A: A job pays 15 dollars per hour. Scenario B: A job pays 12 dollars per hour plus a 30 dollar sign-up fee. Which relationship is non-proportional?
Scenario A
Scenario B
Both
Neither
Explanation
Proportional situations start at zero and scale by a constant rate ($y=kx$). Scenario A has no starting fee, so it passes through the origin and is proportional. Scenario B includes a fixed fee (a $y$-intercept $b\neq 0$), so it is non-proportional.
Two tables relate $x$ to $y$. Table 1: x = 1,2,3,4 and y = 5,9,13,17. Table 2: x = 1,2,3,4 and y = 2,4,6,8. Which table shows a proportional relationship between $x$ and $y$?
Only Table 1
Neither
Both
Only Table 2
Explanation
In a proportional relationship, the ratio $\frac{y}{x}$ is constant and the line would pass through the origin. Table 1 has $\frac{5}{1},\frac{9}{2},\frac{13}{3},\frac{17}{4}$ which are not equal (it matches $y=4x+1$, so $b\neq0$). Table 2 has constant ratio $\frac{y}{x}=2$ (matches $y=2x$), so it is proportional.
Function P: $y=-4x$. Function Q: $y=-4x+10$. How do these relationships differ?
Both are non-proportional because the slope is negative.
Both are proportional because they are linear.
Function P is proportional ($b=0$), while Function Q is non-proportional ($b\neq 0$).
Function P has a $y$-intercept of 10, while Function Q has no $y$-intercept.
Explanation
Proportional linear functions have the form $y=kx$ and pass through the origin ($b=0$). $y=-4x$ is proportional. $y=-4x+10$ has $b=10\neq0$, so it is non-proportional and its graph has a $y$-intercept at 10.