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The image contains a graph that shows a linear relationship, and the associated questions guide you to explore the relationship between \( x \) and \( y \). The points on the graph include \( (1, 1.5) \) and \( (2, 3) \), with arrows indicating changes in \( x \) and \( y \) (denoted as \( \Delta x \) and \( \Delta y \)).

Here is a step-by-step breakdown of how to approach the questions:

### (a) Use the purple "X" to explore how \( x \) and \( y \) change together in this relationship.
- This instruction encourages you to manipulate the values of \( x \) and observe how the corresponding \( y \) values change.
- Based on the graph, when \( x \) increases by 1 (from \( x = 1 \) to \( x = 2 \)), \( y \) increases from \( 1.5 \) to \( 3 \). So, as \( x \) changes by \( +1 \), \( y \) changes by \( +1.5 \).

### (b) Determine the value of the constant ratio \( \Delta y / \Delta x \).
- You need to calculate the slope, which is the ratio of the change in \( y \) to the change in \( x \).
\[
\Delta y = 3 - 1.5 = 1.5 \quad \text{and} \quad \Delta x = 2 - 1 = 1
\]
Therefore, the slope (constant ratio) is:
\[
\frac{\Delta y}{\Delta x} = \frac{1.5}{1} = 1.5
\]

### (c) Therefore, the constant rate of change of \( y \) with respect to \( x \) is:
- The constant rate of change is the same as the slope calculated above, which is \( 1.5 \).

### (d) Define a formula that represents the value of \( y \) in terms of \( x \).
- The equation of a line in slope-intercept form is:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept. From the graph, we can see that when \( x = 0 \), \( y \approx 0 \). Therefore, the y-intercept \( b \) is approximately 0. Thus, the equation becomes:
\[
y = 1.5x
\]

Let me know if you'd like more details on any part of this solution! Here are five related questions to expand on the concepts:

1. How do you find the y-intercept when it's not clearly marked on the graph?
2. What is the importance of the slope in determining the relationship between two variables?
3. How would the graph change if the slope was negative?
4. How can you use a linear equation to predict future values of \( y \)?
5. How does the formula \( y = mx + b \) generalize to non-linear relationships?

**Tip:** Always check the units on both axes of the graph, as they can affect the interpretation of the slope!

The given graph shows a linear relationship between x and y, with the points (1, 1.5) and (2, 3) labeled. Questions include calculating the slope (Δy/Δx), determining the constant rate of change, and defining a formula that represents y in terms of x.

Explore how to calculate the slope and constant rate of change from a graph representing a linear relationship between x and y. Includes step-by-step instructions for determining the equation of the line in the form y = mx + b.

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