Q. What is the y-intercept of \( f(x) = 3x + 2 \)?

Finding the (y)-intercept

To find the (y)-intercept of a linear function, we must understand its position on a coordinate plane. The (y)-intercept is the specific point where the line crosses the vertical (y)-axis.

1. Understand the Concept

   At any point on the (y)-axis, the horizontal position is zero. This means that for any function (f(x)), the (y)-intercept always occurs when (x = 0). To find the value, we simply need to substitute zero for every (x) in the equation.

2. Identify the Given Function

   The function provided is a linear equation in slope-intercept form:

   \(f(x) = 3x + 2\)

   In this form, \(f(x)\) represents the output value (the \(y\)-coordinate) for any given input \(x\).

3. Substitute \(x = 0\) into the Equation

   To locate the intercept, we replace the variable \(x\) with the value \(0\):

   \(f(0) = 3(0) + 2\)

4. Perform the Multiplication

   According to the order of operations, we perform the multiplication first. Any number multiplied by zero results in zero:

   \(3 \cdot 0 = 0\)

   Now the equation looks like this:

   \(f(0) = 0 + 2\)

5. Simplify the Final Value

   Adding \(2\) to \(0\) gives us the final result for the function at that point:

   \(f(0) = 2\)

6. State the Conclusion

   The \(y\)-intercept value is \(2\). When writing this as an ordered pair \((x, y)\), we combine our input of \(0\) with our result of \(2\).

Final Answer:

The (y)-intercept is (2), which corresponds to the point ((0, 2)) on the graph.