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Can you give me the sequences and functions

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:

\[
y = \frac{1}{3}x + \frac{5}{3}
\]

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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.

Use the graph to fill in an answer to the sentence below and find an explicit formula that matches the graph.

Learn how to find the equation of a line using the graph of points. This problem covers linear equations in slope-intercept form, calculating slope, and identifying the y-intercept. Suitable for students in Grades 8-10.

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