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The price for footballs is displayed in the graph. Derive the equation of the proportional relationship through the origin.

To derive the equation of the proportional relationship from the graph, we need to analyze the data points and use the fact that the relationship is linear and passes through the origin $(0, 0)$.

### Step 1: Identify two clear points on the graph.

From the graph, we can extract the following points:
- $(0, 0)$
- $(1, 9)$
- $(2, 18)$
- $(3, 27)$
- $(4, 36)$

### Step 2: Calculate the slope of the line.
The slope $m$ of a linear relationship can be calculated as:

\[
m = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}
\]

Using the points $(0, 0)$ and $(1, 9)$, we get:

\[
m = \frac{9 - 0}{1 - 0} = \frac{9}{1} = 9
\]

So, the slope of the line is 9.

### Step 3: Write the equation.
Since the line passes through the origin, the equation of the line is:

\[
y = mx
\]

Substituting the slope $m = 9$:

\[
y = 9x
\]

Thus, the equation of the proportional relationship is:

\[
\boxed{y = 9x}
\]

This equation indicates that the cost of footballs is proportional to the quantity, with a unit price of 9 dollars per football.

Would you like to explore more details, or have any questions?

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Here are 5 related questions to expand the topic:
1. What would the cost be for 7 footballs based on the equation?
2. How would the equation change if the price per football was increased to $10?
3. How do you determine whether a relationship is proportional from a graph?
4. What is the significance of the graph passing through the origin?
5. Can we use this equation to predict the cost of fractional quantities of footballs?

**Tip:** In proportional relationships, the ratio of $y$ to $x$ (or cost to quantity) remains constant throughout the graph!

Analyze the cost of footballs as shown in the graph and learn how to derive the equation of the proportional relationship through the origin. This problem involves understanding linear equations, proportional relationships, and calculating the slope of a line. Suitable for students in Grades 6-8.

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