Question

which shows a proportional relationship represented by the equation $y = \frac{1}{2}x$ ? (four graphs are shown as options, with coordinate grids and lines. the first graph has a line from (0,0) with a point at (1,2)? wait, no, looking at the grids: first graph: x-axis 0-6, y-axis 0-4, line through (0,0) and a point at (1,2)? wait, no, the first graphs x=1? wait, no, the grids are probably with x and y in integers. wait, the equation is y=(1/2)x. so for x=2, y=1? wait, no, x=2, y=1? wait, no, y=(1/2)x, so x=2, y=1? wait, no, x=2, y=1? wait, no, lets check the graphs. the second graph: x=2, y=1? wait, the second graphs point is at (2,1)? wait, the users image: first graph: line from (0,0), point at (1,2)? no, maybe x=1, y=2? wait, no, the equation is y=(1/2)x, so slope 1/2. so the correct graph should have slope 1/2, passing through origin (since proportional, y=kx, so (0,0) is on it). so the options: first graph: slope 2 (from (0,0) to (1,2)), second: slope 1/2 (from (0,0) to (2,1)? wait, the second graphs point is at (2,1)? wait, the ocr text is: \which shows a proportional relationship represented by the equation y = ½x ?\ with four graphs: first graph: line from (0,0), point at (1,2) (slope 2), second: line from (0,0), point at (2,1) (slope 1/2), third: line with y-intercept 2 (not proportional), fourth: line with y-intercept (not proportional). so the question is to choose the graph with y=(1/2)x, which is a proportional relationship (passes through origin, slope 1/2).

Explanation:

Step1: Recall proportional relationship

A proportional relationship \( y = kx \) (where \( k \) is the constant of proportionality) has a graph that passes through the origin \((0,0)\) and has a slope of \( k \). Here, \( k=\frac{1}{2} \), so the slope is \( \frac{1}{2} \) and it should pass through \((0,0)\).

Step2: Analyze each graph

• Top - left graph: Let's check the slope. At \( x = 1 \) (assuming grid is 1 unit per square), \( y = 2 \). Slope \(=\frac{y}{x}=\frac{2}{1}=2

eq\frac{1}{2}\).

• Top - right graph: At \( x = 2 \), \( y = 1 \) (since \( y=\frac{1}{2}(2)=1 \)). Slope \(=\frac{1}{2}\), and it passes through \((0,0)\).
• Bottom - left graph: It has a \( y \) - intercept at \( y = 2 \), so it's not proportional (proportional graphs pass through \((0,0)\)).
• Bottom - right graph: It has a \( y \) - intercept (starts above \( (0,0) \)), so not proportional.

Answer:

The top - right graph (the second graph in the top row) represents the proportional relationship \( y=\frac{1}{2}x \).