Problem 18 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line with slope \(\frac{1}{2}\) and \(y\) intercept \(-1\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Slope and Intercept

The slope of the line is given as \( \frac{1}{2} \), and the \( y \)-intercept is \(-1\). This information allows us to use the equation of a line in slope-intercept form.

Step 2: Use Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Here, \( m = \frac{1}{2} \) and \( b = -1 \). So, the equation becomes \( y = \frac{1}{2}x - 1 \).

Step 3: Sketch the Line

To sketch the line, start at the \( y \)-intercept, which is \(-1\) on the \( y \)-axis. Then use the slope \( \frac{1}{2} \) to determine the next point: rise 1 unit up and run 2 units to the right. Plot this second point.

Step 4: Draw the Line

Once you have at least two points, draw a straight line through them extending in both directions, which represents the infinite nature of a linear equation. Make sure the line crosses the \( y \)-axis at \(-1\) and follows the plotted slope.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a method of expressing the equation of a straight line using two main components: the slope and the y-intercept. This form is written as:
\[ y = mx + b \]
Here,

• \( y \) is the dependent variable (usually represented on the vertical axis of a graph).
• \( m \) represents the slope of the line.
• \( x \) is the independent variable (usually represented on the horizontal axis of a graph).
• \( b \) is the y-intercept.

When given a slope and a y-intercept, you can directly plug these values into the slope-intercept form to get the equation of the line.
This makes it easy to define a line and is especially helpful for graphing and solving problems involving linear equations.

Graphing Lines

Graphing a line involves plotting points on a coordinate plane and connecting them to form a straight line. To graph a line in slope-intercept form:

• Start by identifying the y-intercept \( b \). This point is where the line crosses the y-axis.
• Plot the y-intercept on the coordinate plane.
• Use the slope \( m \) to determine another point on the line. Remember, slope is the ratio of "rise" over "run."
• "Rise" refers to how far up or down you go from the y-intercept, and "run" refers to how far right you go.

Once you have at least two points, draw a straight line through them. Extend the line in both directions to indicate it continues indefinitely. Graphing lines visually represents the relationship between variables and can help confirm calculations for linear equations.

Slope of a Line

The slope of a line measures its steepness and direction on a graph. It is represented by \( m \) in the slope-intercept form.
The slope is calculated as the ratio of the change in the y-values (rise) to the change in the x-values (run) between two points on the line:
\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1} \]

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope indicates a vertical line.

Understanding slope is crucial for interpreting the behavior of a line and predicting values along its path.

y-intercept

The y-intercept of a line is the point where it crosses the y-axis. It is denoted by \( b \) in the slope-intercept form.
To find the y-intercept in the slope-intercept equation \( y = mx + b \), simply look at the value of \( b \). This is the y-coordinate of the point on the line where \( x = 0 \).
For example, in the equation \( y = \frac{1}{2}x - 1 \):

• The y-intercept is \(-1\).

This means the line will cross the y-axis at the point (0, -1).
The y-intercept is helpful when graphing, as it provides a fixed starting point from which the slope can be used to locate additional points on the line.