Find the linear function whose graph passes through the point (3, 2) and is parallel to the line $y=3x+8$ .

Verified step by step guidance

1. Identify the slope of the given line, which is parallel to the line we need to find. The given line is \( y = 3x + 8 \), so the slope \( m \) is 3.
2. Use the point-slope form of a linear equation, which is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
3. Substitute the point \( (3, 2) \) and the slope \( m = 3 \) into the point-slope form: \( y - 2 = 3(x - 3) \).
4. Simplify the equation to get it into the slope-intercept form \( y = mx + b \).
5. Distribute the 3 on the right side: \( y - 2 = 3x - 9 \), then add 2 to both sides to solve for \( y \).

Key Concepts

Linear Functions

A linear function is a mathematical expression that describes a straight line when graphed. It is typically represented in the form y = mx + b, where m is the slope and b is the y-intercept. Understanding linear functions is crucial for determining the relationship between variables and for solving problems involving lines.

Slope

The slope of a line measures its steepness and direction, calculated as the change in y divided by the change in x (rise over run). For two parallel lines, the slopes are equal. In this question, the slope of the given line y = 3x + 8 is 3, which will be the same for the linear function we need to find.

Point-Slope Form

The point-slope form of a linear equation is expressed as y - y1 = m(x - x1), where (x1, y1) is a specific point on the line and m is the slope. This form is particularly useful for writing the equation of a line when you know a point on the line and its slope, allowing for straightforward calculations in this problem.