which statement about proportional relationships is false?\na proportional relationship must graph as a…

Explanation:

Step1: Recall properties of proportional relationships

A proportional relationship is of the form $y = kx$, where $k$ is the constant of proportionality. Its graph is a straight - line passing through the origin $(0,0)$ and for any two points $(x_1,y_1)$ and $(x_2,y_2)$ in the relationship, $\frac{y_1}{x_1}=\frac{y_2}{x_2}=k$.

Step2: Analyze each option

• Option A: The equation of a proportional relationship $y = kx$ is a linear equation of the form $y=mx + b$ where $m = k$ and $b = 0$. The graph of a linear equation $y=mx + b$ is a straight - line. So, a proportional relationship must graph as a straight line. This statement is true.
• Option B: Substitute $x = 0$ into $y=kx$. We get $y=k\times0 = 0$. So, the graph of a proportional relationship must pass through $(0,0)$. This statement is true.
• Option C: If $(x_1,y_1)$ and $(x_2,y_2)$ are points in a proportional relationship $y = kx$, then $\frac{y_1}{x_1}=k$ and $\frac{y_2}{x_2}=k$. So, each point (or pair) in a proportional relationship must share the same ratio. This statement is true.
• Option D: In a proportional relationship, the ratio between $y$ and $x$ is constant ($\frac{y}{x}=k$), not the difference. For example, in the proportional relationship $y = 2x$, if $x_1=1,y_1 = 2$ and $x_2 = 2,y_2=4$. The ratio $\frac{y_1}{x_1}=\frac{2}{1}=2$ and $\frac{y_2}{x_2}=\frac{4}{2}=2$, but the difference $y_2 - y_1=4 - 2 = 2$ and $x_2 - x_1=2 - 1 = 1$, and $\frac{y_2 - y_1}{x_2 - x_1}\neq\frac{y_1}{x_1}$. This statement is false.

Answer:

D. Each point (or pair) in a proportional relationship must share the same difference.