Problem 29 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ y=\frac{x+2}{3} $$

Step-by-Step Solution

Answer

Slope \( m = \frac{1}{3} \), \( y \)-intercept \((0, \frac{2}{3})\).

Step 1: Identify the Form of the Equation

The equation given is \( y = \frac{x+2}{3} \). To identify the slope and \( y \)-intercept, we need to recognize that this is a linear equation. The standard form for a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.

Step 2: Rearrange the Equation

Rearrange \( y = \frac{x+2}{3} \) into the form \( y = mx + b \). We have \( y = \frac{1}{3}x + \frac{2}{3} \), where \( \frac{1}{3} \) is the coefficient of \( x \) and \( \frac{2}{3} \) is the constant term.

Step 3: Determine the Slope

From the rearranged equation \( y = \frac{1}{3}x + \frac{2}{3} \), the slope \( m \) is identified as \( \frac{1}{3} \).

Step 4: Determine the Y-Intercept

The \( y \)-intercept \((0, b)\) is the point where the line crosses the \( y \)-axis. From the equation \( y = \frac{1}{3}x + \frac{2}{3} \), the \( y \)-intercept is \( b = \frac{2}{3} \). Therefore, the point is \( (0, \frac{2}{3}) \).

Step 5: Graph the Line

Start by plotting the \( y \)-intercept \( (0, \frac{2}{3}) \) on the graph. Use the slope \( \frac{1}{3} \) (which means for every 1 unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \)) to find another point. For example, going from \( (0, \frac{2}{3}) \) to \( (3, 1) \) indicates a rise of 1 over a run of 3. Draw a line through these points extending in both directions.

Key Concepts

Slope

The slope of a line tells you how steep the line is. It is a way to describe how rapidly the line is moving upward or downward as you move along it. For any line in the equation form of \( y = mx + b \), the slope is denoted by the letter \( m \). The slope can be calculated by the ratio of the change in \( y \) over the change in \( x \) (often referred to as "rise over run").

• If the slope is positive, the line will ascend from left to right, meaning the line rises as it moves from left to right.
• If negative, it descends, meaning the line falls.
• A slope of zero is a horizontal line, indicating no rise as you move along.
• An undefined slope corresponds to a vertical line, where there's no run.

For example, in the equation \( y = \frac{1}{3}x + \frac{2}{3} \), the slope \( m \) is \( \frac{1}{3} \). This means that for every increase of 3 units along the \( x \)-axis, the line rises by 1 unit on the \( y \)-axis.

Y-Intercept

The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. This is represented as a coordinate \( (0, b) \), where \( b \) is the \( y \)-intercept found in the equation \( y = mx + b \).

• The \( y \)-intercept can be thought of as the starting value or the point on the \( y \)-axis when the \( x \) value is zero.
• It is the constant term in the equation of a line in slope-intercept form.

In our example of the equation \( y = \frac{1}{3}x + \frac{2}{3} \), the \( y \)-intercept \( b \) is \( \frac{2}{3} \). This means at the point \((0, \frac{2}{3})\), this is where the line crosses the \( y \)-axis.

Graphing Linear Equations

Graphing a linear equation involves plotting points on a coordinate plane and then connecting them to form a line. To graph using the slope-intercept form \( y = mx + b \), follow these steps:

• Start by plotting the \( y \)-intercept \( (0, b) \) on the graph. This is the point where the line touches the \( y \)-axis, making it a natural starting point.
• Utilize the slope \( m \) to determine the next point. The slope indicates the direction and steepness of the line.
• For a slope of \( \frac{1}{3} \), move 1 unit up or down for every 3 units you move horizontally.
• Plot this second point based on your slope calculation, then draw a line through these points extending in both directions.

The process ensures that your line accurately represents the equation and reflects the slope and intercept effectively. For the equation \( y = \frac{1}{3}x + \frac{2}{3} \), you start at \( (0, \frac{2}{3}) \) and use the slope to find another point, such as \( (3, 1) \), then connect them.

Standard Form of a Line

The standard form of a linear equation is another way to express a line's equation. It's different from the slope-intercept form \( y = mx + b \). In standard form, a line is expressed as \( Ax + By = C \), where \( A, B, \) and \( C \) are integers. Some key points about the standard form are:

• It is useful in situations where you need to identify or solve for each variable individually.
• To convert from slope-intercept form \( y = mx + b \) to standard form, rearrange the terms to isolate \( 0 \) on one side.
• This can involve multiplying to clear fractions for a tidy integer format.
• In the example equation \( y = \frac{1}{3}x + \frac{2}{3} \), converting it into standard form would involve eliminating fractions and reordering, possibly resulting in \( x - 3y = -2 \) after clearing fractions by multiplying through by the denominator.

This form can seem abstract but provides a different yet helpful perspective on understanding lines.