Problem 44 - Calculus Single Variable

Question

Sketch the line whose Cartesian equation is given. $$ y=(x-2) / 2+4 $$

Step-by-Step Solution

Answer

The line has a slope of 1/2 and crosses the y-axis at (0, 3).

Step 1: Understand the equation

The given equation is in the form \( y = \frac{x - 2}{2} + 4 \). This is a linear equation, which describes a straight line in the coordinate plane.

Step 2: Simplify the equation

Simplify the equation by distributing the division: \( y = \frac{1}{2}x - 1 + 4 \). Simplify further to \( y = \frac{1}{2}x + 3 \), representing it in the slope-intercept form \( y = mx + b \).

Step 3: Identify slope and y-intercept

In the equation \( y = \frac{1}{2}x + 3 \), the slope \( m \) is \( \frac{1}{2} \), and the y-intercept \( b \) is 3. This means the line crosses the y-axis at 3 and has a slope of \( \frac{1}{2} \).

Step 4: Plot the y-intercept

Begin by plotting the y-intercept point (0, 3) on the coordinate plane, marking where the line will cross the y-axis.

Step 5: Use the slope to find another point

From the y-intercept, use the slope \( \frac{1}{2} \) to find another point. A slope of \( \frac{1}{2} \) means rise 1 unit and run 2 units. Starting at (0, 3), move up 1 unit and right 2 units to reach the point (2, 4).

Step 6: Draw the line

With the points (0, 3) and (2, 4) plotted, draw a straight line through them. This line is the graphical representation of the equation \( y = \frac{x - 2}{2} + 4 \).

Key Concepts

Slope-Intercept Form

To truly understand the equation from the exercise, we must first dive into the concept of the slope-intercept form. This form, expressed as \( y = mx + b \), is a linear equation where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept.

This form is popular because it immediately gives us the slope and the y-intercept, making it convenient to graph the line.
The slope tells us how steep the line is, indicating the angle at which it inclines or declines. The y-intercept is the value where the line crosses the y-axis, acting as a starting point for graphing.
By converting the given equation \( y = \frac{x - 2}{2} + 4 \) into the slope-intercept form \( y = \frac{1}{2}x + 3 \), we can see

• the slope \( m \) is \( \frac{1}{2} \)
• the y-intercept \( b \) is 3

Understanding the slope-intercept form is crucial for identifying key features of the line quickly.

Linear Equations

Linear equations, such as the one given in this exercise, represent straight lines on a graph. They are key in algebra and often appear in a variety of scientific and engineering contexts.
Typically written in the form \( ax + by = c \) or the slope-intercept form \( y = mx + b \), these equations are characterized by having constant or straight-line relationships between two variables.
For every change in the x-value, indicated by the slope \( m \), there will be a proportional change in the y-value. This proportional change is consistent, making the graph a straight line.
Linear equations have infinite solutions since they describe a line that continues without end in both directions. They can model numerous real-world situations where a constant rate of change is present, such as speed or density.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where each point is determined by a pair of numbers. These numbers, known as coordinates, are typically represented as \((x, y)\).
The plane is defined by a horizontal axis, the x-axis, and a vertical axis, the y-axis, intersecting at a point called the origin \((0, 0)\).
In our exercise, the coordinate plane is used to graph the line defined by the linear equation \( y = \frac{1}{2}x + 3 \).
To plot the line, we first identify the y-intercept and plot it on the y-axis. From this point, we apply the slope \( \frac{1}{2} \), indicating a rise of 1 unit for every run of 2 units, to determine additional points.

• Start at the y-intercept (0, 3).
• Move up 1 unit and right 2 units to plot the second point (2, 4).

The simplicity of the coordinate plane allows us to visualize relationships described by linear equations easily. It is a fundamental tool in geometry and algebra for understanding spatial relationships.