Problem 22 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=\frac{1}{2}, b=-\frac{1}{3} $$

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{2}x - \frac{1}{3} \).

Step 1: Recognize the Equation Form

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Given Slope

Substitute the given slope, \( m = \frac{1}{2} \), into the general slope-intercept form equation. This yields: \( y = \frac{1}{2}x + b \).

Step 3: Substitute the Given Y-Intercept

Substitute the given y-intercept, \( b = -\frac{1}{3} \), into the equation from Step 2. This results in: \( y = \frac{1}{2}x - \frac{1}{3} \).

Key Concepts

Understanding the Slope

The "slope" of a line is a measure of its steepness and direction. It indicates how much the line rises or falls as you move from left to right along the graph. The slope is often denoted by the letter \( m \). In mathematical terms, the slope is calculated as the "rise over run". This means the vertical change (up or down) divided by the horizontal change (left or right) between two points on the line. For example:

• If a line rises as it moves from left to right, it has a positive slope.
• If it falls, the slope is negative.
• A flat horizontal line has a slope of zero, as it does not rise or fall.
• A vertical line, which isn't defined by slope-intercept form, has an undefined slope.

The slope in the equation \( y = mx + b \) signifies how tilted the line is. In our example, the slope \( m \) is \( \frac{1}{2} \), meaning for every unit moved horizontally, the line moves up by half a unit.

The Role of the Y-Intercept

The "y-intercept" is the point where the line crosses the y-axis on a graph. This point is crucial because it allows us to see how the line behaves when it starts. The y-intercept is represented by the letter \( b \) in the slope-intercept form equation \( y = mx + b \).
To find the y-intercept on a graph, look where the line meets the y-axis. This point will have coordinates \((0, b)\); where \( b \) is the value of the y-intercept. In simple terms, it's the value of \( y \) when \( x \) is zero.

• If the y-intercept is positive, the line will cross above the origin (where the x and y axes intersect).
• If negative, it will cross below the origin.
• If \( b = 0 \), the line passes directly through the origin.

In our specific example, the y-intercept is \( -\frac{1}{3} \), meaning the line crosses the y-axis at -1/3.

Relating Linear Equations

"Linear equations" are mathematical expressions that create straight lines when graphed. The slope-intercept form of a linear equation, \( y = mx + b \), is the most commonly used form to quickly graph or understand these equations due to its clear representation of slope and y-intercept.

• \( y = mx + b \) shows all lines of different slopes (\( m \)) will have unique angles.
• Different y-intercepts (\( b \)) determine where these lines start on the y-axis.
• Linear equations can be rearranged into other formats, but slope-intercept is popular for its clarity.

Linear equations form the basis for more complex functions. Understanding them is key to mastering more advanced mathematical concepts. Our equation \( y = \frac{1}{2}x - \frac{1}{3} \) perfectly illustrates the interaction between slope, y-intercept, and linearity.