From the graph in the image, we are tasked with finding the rate of change and the initial value of the linear function.

Step 1: Identify two points on the line

From the graph, the line passes through the points:

• $(0, 6)$ (the y-intercept)
• $(4, 0)$

Step 2: Calculate the rate of change (slope)

The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute $(x_1, y_1) = (0, 6)$ and $(x_2, y_2) = (4, 0)$: $m = \frac{0 - 6}{4 - 0} = \frac{-6}{4} = -\frac{3}{2}$

Thus, the rate of change (slope) is: $m = -\frac{3}{2}$

Step 3: Identify the initial value

The initial value is the $y$-intercept, which occurs when $x = 0$. From the graph, the $y$-intercept is $6$.

Final Answer:

• Rate of change: $-\frac{3}{2}$
• Initial value: $6$

Would you like more details on how this was calculated, or any further explanations?

Here are 5 related questions to expand your learning:

1. How do you find the slope of a line given any two points?
2. What is the equation of this line in slope-intercept form ($y = mx + b$)?
3. How can you determine the rate of change from a table of values instead of a graph?
4. What does a negative slope represent in a real-world context?
5. How do you find the initial value if the graph does not include the $y$-intercept?

Tip: Always double-check your points before calculating the slope to avoid sign errors!