Find the equation of the given linear function

To find the equation of the given linear function, we'll use the given points \( (x, f(x)) \):

- \( (1, 6) \)
- \( (2, 9) \)
- \( (3, 12) \)
- \( (4, 15) \)

### Step 1: Calculate the slope (\( m \)).
The slope \( m \) of a linear function is determined by the formula:
\[
m = \frac{\Delta f(x)}{\Delta x}
\]
Using any two points, for example, \( (1, 6) \) and \( (2, 9) \):
\[
m = \frac{9 - 6}{2 - 1} = \frac{3}{1} = 3
\]

### Step 2: Determine the equation of the linear function \( f(x) \).
The general form of a linear function is:
\[
f(x) = mx + b
\]
Substitute \( m = 3 \) into the equation:
\[
f(x) = 3x + b
\]

### Step 3: Solve for the y-intercept (\( b \)).
Use one of the given points, for example, \( (1, 6) \):
\[
6 = 3(1) + b
\]
\[
6 = 3 + b \implies b = 3
\]
Thus, the equation of the function is:
\[
f(x) = 3x + 3
\]

### Final Result:
The equation of the given linear function is \( f(x) = 3x + 3 \).

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How can you verify if the equation \( f(x) = 3x + 3 \) is correct for all given points?
2. What is the significance of the slope in this linear function?
3. How would the equation change if the points had a different pattern?
4. Can we determine the x-intercept from the equation \( f(x) = 3x + 3 \)?
5. How would you graph this function on a coordinate plane?

**Tip:** When finding the equation of a line, always double-check by substituting the points back into the equation to ensure accuracy.

Learn how to find the equation of a linear function given specific points. This detailed guide covers calculating the slope, using the slope-intercept form, and determining the y-intercept. Suitable for Grades 7-9 students.

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