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.