The graph shows the equation representing Relationship A. We can calculate the slope (rate of change) for Relationship A by using the two points given on the line: $(4, 2)$ and $(8, 4)$.

The slope $m$ is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points:

$m = \frac{4 - 2}{8 - 4} = \frac{2}{4} = \frac{1}{2}$

Thus, the equation for Relationship A is:

$y = \frac{1}{2}x$

Since the question states that Relationship B has a greater rate (slope) than Relationship A, we are looking for equations where the slope is greater than $\frac{1}{2}$.

Let's analyze the given options:

1. $y = \frac{3}{4}x$ → Slope = $\frac{3}{4}$, which is greater than $\frac{1}{2}$.
2. $y = \frac{2}{3}x$ → Slope = $\frac{2}{3}$, which is greater than $\frac{1}{2}$.
3. $y = \frac{1}{4}x$ → Slope = $\frac{1}{4}$, which is less than $\frac{1}{2}$.
4. $y = 0.6x$ → Slope = $0.6 = \frac{6}{10} = \frac{3}{5}$, which is greater than $\frac{1}{2}$.

Correct answers:

• $y = \frac{3}{4}x$
• $y = \frac{2}{3}x$
• $y = 0.6x$

Would you like further details on how to interpret the slope and its impact on graph steepness?

Here are 5 related questions:

1. How do you calculate the slope of a line given two points?
2. What does the slope represent in a linear equation?
3. How does the steepness of the line relate to the slope?
4. Can you explain the difference between positive and negative slopes?
5. How would a vertical or horizontal line's equation differ in terms of slope?

Tip: In a linear equation $y = mx + b$, the slope $m$ determines the steepness of the line, while $b$ (the y-intercept) determines where the line crosses the y-axis.