Question

m3|l2 graphs of proportional relationships now, let’s consider graph 2. what’s the equation for graph 2? use x to represent the time in hours, and use y to represent the rainfall in inches. graph 2 (rainfall (in) axis from 0 to 2.0, time axis not fully shown, grid with a line from origin to a point around (x, 0.4) maybe)

Explanation:

Step1: Identify proportional relationship

Proportional relationships have the form \( y = kx \), where \( k \) is the constant of proportionality (slope). The graph passes through the origin \((0,0)\), so it's a proportional relationship (no \( y \)-intercept other than 0).

Step2: Find the slope \( k \)

From the graph, we can pick a point. Let's assume a point like when \( x \) (time in hours) is, say, 4 hours (since the grid likely has \( x \)-axis units), and \( y \) (rainfall) is 0.4 inches? Wait, no, looking at the graph, when \( x \) is, let's see the grid. Wait, the \( y \)-axis is rainfall (in) and \( x \)-axis is time (hours). Let's take the point where \( y = 0.4 \) inches. Let's find \( x \) for that. Wait, maybe the slope is \( k=\frac{y}{x} \). Let's see, if we take a point, say when \( x = 4 \), \( y = 0.4 \)? Wait, no, maybe the slope is \( 0.1 \)? Wait, no, let's calculate. Wait, the options have 0.1 and 0.4. Wait, let's check the graph. The line goes through the origin. Let's take two points: (0,0) and, say, (4, 0.4). Then slope \( k=\frac{0.4}{4}=0.1 \). Wait, no, \( \frac{y}{x}=k \), so \( y = kx \). Wait, if \( x = 4 \), \( y = 0.4 \), then \( k = \frac{0.4}{4}=0.1 \)? No, \( 0.4\div4 = 0.1 \)? Wait, 4 times 0.1 is 0.4? Wait, no, 4 times 0.1 is 0.4? Wait, 0.1 times 4 is 0.4? Yes. Wait, but the equation is \( y = kx \), so \( k = \frac{y}{x} \). So if \( x = 4 \), \( y = 0.4 \), then \( k = 0.4/4 = 0.1 \)? Wait, no, 0.4 divided by 4 is 0.1? Wait, 4 0.1 = 0.4. So \( y = 0.1x \)? Wait, no, wait, the problem says use \( x \) for time, \( y \) for rainfall. Wait, maybe I got \( x \) and \( y \) reversed. Wait, the graph's y-axis is rainfall (in), x-axis is time (hours). So the relationship is rainfall (y) vs time (x). So the slope is \( \frac{y}{x} = k \), so \( y = kx \). Let's take a point: suppose when \( x = 4 \) hours, \( y = 0.4 \) inches. Then \( k = 0.4 / 4 = 0.1 \)? Wait, no, 0.4 divided by 4 is 0.1? Wait, 4 0.1 = 0.4. So \( y = 0.1x \)? Wait, no, that would mean when \( x = 4 \), \( y = 0.4 \), which matches. Wait, but the options have \( y = 0.1x \) or \( y = 0.4x \)? Wait, no, the buttons are \( \cdot \), \( x \), \( = \), \( 0.1 \), \( y \), \( + \), \( 0.4 \). Wait, maybe the equation is \( y = 0.1x \)? Wait, no, let's re-express. Wait, the problem is to find the equation. So proportional relationship: \( y = kx \). Let's check the slope. Let's take \( x = 1 \), then \( y = 0.1 \)? Or \( x = 4 \), \( y = 0.4 \). So \( k = 0.4 / 4 = 0.1 \). So the equation is \( y = 0.1x \), which can be written as \( 0.1x = y \) or \( y = 0.1x \). Wait, the buttons are \( \cdot \), \( x \), \( = \), \( 0.1 \), \( y \), \( + \), \( 0.4 \). So to form the equation, we need to arrange \( y = 0.1x \), which is \( 0.1x = y \) or \( y = 0.1x \). Wait, maybe the correct equation is \( y = 0.1x \), so \( 0.1 \times x = y \), which is \( 0.1x = y \) or \( y = 0.1x \).

Wait, maybe I made a mistake. Let's re-express. The proportional relationship is \( y = kx \), where \( k \) is the constant of proportionality. From the graph, when \( x = 4 \) (time in hours), \( y = 0.4 \) (rainfall in inches). Then \( k = \frac{y}{x} = \frac{0.4}{4} = 0.1 \). So \( y = 0.1x \), which can be written as \( 0.1x = y \) (since \( y = 0.1x \) is equivalent to \( 0.1x = y \)).

Answer:

\( y = 0.1x \) (or \( 0.1x = y \))