Problem 23 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y=\frac{1}{2} x+1 $$

Step-by-Step Solution

Answer

Plot (0, 1), use the slope 1/2 to plot (2, 2), and draw the line.

Step 1: Identify the Type of Equation

The given equation is in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the equation is \(y = \frac{1}{2}x + 1\).

Step 2: Interpret the Slope and Y-Intercept

The slope \(m\) is \(\frac{1}{2}\), which means for every 2 units you move to the right on the x-axis, you move 1 unit up on the y-axis. The y-intercept \(b\) is 1, indicating that the line crosses the y-axis at the point (0,1).

Step 3: Plot the Y-Intercept

Begin by plotting the point where the line crosses the y-axis. From the equation \(y = \frac{1}{2}x + 1\), the y-intercept is 1, so place a point at (0, 1).

Step 4: Use the Slope to Find Another Point

Starting from the y-intercept (0, 1), use the slope \(\frac{1}{2}\) to find another point. Move 2 units to the right (increase x by 2) and 1 unit up (increase y by 1) to reach the new point (2, 2).

Step 5: Draw the Line

Use the two points, (0, 1) and (2, 2), to draw a straight line across the graph. This line is the visual representation of the equation \(y = \frac{1}{2}x + 1\).

Step 6: Check Your Line

Verify that the line represents the equation by ensuring it aligns with the slope \(\frac{1}{2}\) and crosses the y-axis at 1. Optionally, choose another point on the line to see if it satisfies the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward method to express a linear equation. The general structure is written as: \[ y = mx + b \] where \( m \) represents the slope of the line, and \( b \) is the y-intercept.
This form makes it easy to identify key features of the line just by looking at the equation.

• y: Represents the y-coordinate on the graph.
• x: Represents the x-coordinate on the graph.
• m: The slope, indicating the steepness and direction of the line.
• b: The y-intercept, or the point where the line crosses the y-axis.

Using slope-intercept form, one can easily graph a line by plotting the y-intercept first and then using the slope to determine the direction and angle of the line. This is why many find the slope-intercept form to be very intuitive for graphing purposes.

Slope

The slope is crucial as it reveals how steep a line is and which direction it goes. Given a linear equation in slope-intercept form \( y = mx + b \), we identify the slope as \( m \).
For the equation \( y = \frac{1}{2}x + 1 \), the slope \( m \) is \( \frac{1}{2} \). This tells us:

• The line rises 1 unit vertically for every 2 units it moves horizontally to the right.
• If the slope were negative, the line would decrease instead of rise.

To visualize it better: starting from a point on the line, move 2 units right for 1 unit up to find the next point of the line. Lines with greater slopes are steeper. Conversely, smaller slopes make a more gradual incline, and zero slopes create horizontal lines.

Y-Intercept

The y-intercept of a line is the spot where it crosses the y-axis, making it a pivotal point for graphing linear equations. In the slope-intercept form \( y = mx + b \), the term \( b \) is the y-intercept.
For the equation \( y = \frac{1}{2}x + 1 \), the y-intercept is identified as \( 1 \). This means:

• The line will cross the y-axis at the point \( (0, 1) \).

To graph this, you start by plotting this point. From there, using the slope, other points on the line are determined. A clear understanding of the y-intercept helps anticipate where the line is anchored vertically on the graph. It's a primary reference point for any changes or transformations on the line.