The graph shown depicts several points, including $(1, 2)$, $(4, 3)$, and $(7, 4)$. From the coordinates, we can identify a linear pattern since the $y$-values increase at a steady rate as the $x$-values increase.

Finding the explicit formula (linear equation):

We will start by determining the equation of the line passing through the given points. The general form of a linear equation is:

$y = mx + b$

where:

• $m$ is the slope,
• $b$ is the y-intercept.

Step 1: Calculate the slope $m$

The slope $m$ is calculated using the formula:

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

Taking the points $(1, 2)$ and $(7, 4)$:

$m = \frac{4 - 2}{7 - 1} = \frac{2}{6} = \frac{1}{3}$

So, the slope is $m = \frac{1}{3}$.

Step 2: Use the point-slope form to find the y-intercept $b$

We can substitute one of the points, say $(1, 2)$, into the equation $y = mx + b$ to solve for $b$:

$2 = \frac{1}{3}(1) + b$

$2 = \frac{1}{3} + b$

$b = 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3}$

So, the y-intercept $b = \frac{5}{3}$.

Step 3: Write the final equation

The equation of the line that best matches the given points is:

$y = \frac{1}{3}x + \frac{5}{3}$

Conclusion:

The graph represents a linear relationship, and the explicit formula that matches the graph is:

$y = \frac{1}{3}x + \frac{5}{3}$

Would you like more details on any part of this process, or have any specific questions?

Here are 5 related questions to further expand your understanding:

1. How do we determine if a graph represents a linear function?
2. Can you derive the equation of a line given only two points?
3. What is the geometric meaning of the slope and y-intercept in a linear function?
4. How do different slopes affect the steepness of a line?
5. How can we graph a line once we know its equation?

Tip: To verify a linear equation from a graph, always check the slope and one point on the graph.