Problem 27 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ g(x)=\frac{1}{2} x $$

Step-by-Step Solution

Answer

Slope: \( \frac{1}{2} \), y-intercept: \( 0 \).

Step 1: Understand the function

The function given is \( g(x) = \frac{1}{2}x \). This is a linear function in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify the slope and y-intercept

From the function \( g(x) = \frac{1}{2}x \), we can see that \( m = \frac{1}{2} \). There's no constant term, so \( b = 0 \). Thus, the slope is \( \frac{1}{2} \) and the y-intercept is \( 0 \).

Step 3: Plot the y-intercept

Start by plotting the y-intercept on the graph. Since \( b = 0 \), the point (0, 0) is where the line will cross the y-axis.

Step 4: Use the slope to find another point

The slope \( \frac{1}{2} \) tells us that for every 1 unit moved in the x-direction, the function increases by \( \frac{1}{2} \) units in the y-direction. From (0, 0), move 1 unit to the right (x = 1) and \( \frac{1}{2} \) units up (y = \frac{1}{2}) to find another point, (1, \frac{1}{2}).

Step 5: Draw the line

With the points (0, 0) and (1, \frac{1}{2}) plotted, draw a straight line through these points to extend in both directions. This line represents the graph of the function \( g(x) = \frac{1}{2}x \).

Step 6: Verify your graph

Ensure that your graph is a straight line passing through the origin (0, 0) and remember that your line should have a consistent slope of \( \frac{1}{2} \). Check by picking a point on the line other than the ones used and confirm the slope calculation is consistent.

Key Concepts

Slope Identification

When working with linear equations, identifying the slope is crucial. The slope essentially measures the steepness and direction of a line. It's represented by the letter \( m \) in the equation of a line written in the slope-intercept form, \( y = mx + b \). In the function \( g(x) = \frac{1}{2}x \), the slope \( m \) is \( \frac{1}{2} \).

This tells us a lot about the line:

• For every increase of 1 unit in the x-direction (or horizontally), the function increases by \( \frac{1}{2} \) units in the y-direction (or vertically). This characteristic of the slope helps in plotting the graph by making it predictable.
• A positive slope means that as you move to the right, the line goes up. If it were negative, the line would go down as you move to the right.
• Since the slope is a fraction, it can be interpreted as the 'rise' over the 'run'. Thus, the line rises \( \frac{1}{2} \) while it runs 1 unit to the right.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is represented by \( b \) in the equation \( y = mx + b \). For the linear function given, \( g(x) = \frac{1}{2}x \), the y-intercept \( b \) is 0.

This results in a few important implications:

• The line passes through the origin point (0,0) since there is no constant term to shift the line upwards or downwards on the y-axis.
• The y-intercept is essential as it serves as one of the fixed points that guide the line’s path on the graph.
• Even if the line shifts due to changes in the linear equation, the y-intercept gives a starting point for plotting the graph.
• In equations with a y-intercept other than zero, the line would cross the y-axis at that point instead.

Linear Equations

Linear equations represent straight-line graphs in the coordinate plane, and are typically written in the form \( y = mx + b \). This simple structure makes them easy to understand and work with. Here are some clear characteristics and important facts about linear equations:

• The constant \( m \) is the slope of the line. It determines the line's tilt and the direction (upwards for positive, downwards for negative).
• The constant \( b \) is the y-intercept, dictating where the line cuts through the y-axis.
• These equations model a constant rate of change, which makes predicting future trends straightforward from a graph. Such equations are invaluable in various fields like physics, economics, and engineering.
• Basic operations like solving for \( y \), or plotting can start from rearranging the equation into slope-intercept form if it appears in another format.

Drawing and understanding graphs from linear equations become simple once you grasp slope and y-intercept, as they guide exactly how to plot and comprehend each line.