Problem 78 - College Algebra

Question

For the following exercises, sketch the graph of each equation. $$ h(x)=\frac{1}{3} x+2 $$

Step-by-Step Solution

Answer

Graph a straight line through points (0,2) and (3,3) with slope \( \frac{1}{3} \).

Step 1: Identify the Type of Function

The given equation is \( h(x) = \frac{1}{3}x + 2 \). This is a linear function, which is characterized by the equation \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Determine the Slope and Y-intercept

In the equation \( h(x) = \frac{1}{3}x + 2 \), the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is 2. This means the line crosses the y-axis at (0, 2).

Step 3: Plot the Y-intercept

Begin by plotting the y-intercept on the graph. Place a point at (0,2) on the y-axis. This is where the line will cross the y-axis.

Step 4: Use the Slope to Find Another Point

The slope \( \frac{1}{3} \) indicates that for every 3 units you move to the right along the x-axis, the line rises by 1 unit. From the y-intercept (0,2), move right 3 units to (3,2), then up 1 unit to (3,3). Plot the point (3,3).

Step 5: Draw the Line

With points (0,2) and (3,3) plotted, draw a straight line through these points extending in both directions. This line represents the graph of the equation \( h(x) = \frac{1}{3}x + 2 \).

Key Concepts

Graphing Linear Functions

Graphing a linear function involves plotting a straight line on a coordinate plane.
These functions are characterized by their constant slope and y-intercept.
The general form of a linear function is expressed as:

• \( y = mx + b \)

where:

• \( m \) represents the slope.
• \( b \) represents the y-intercept.

To graph a linear function:
You'll need to plot at least two points and draw a line through them.
These steps are quite straightforward, making linear equations one of the simplest forms to work with in algebra.
By determining the slope and y-intercept, you can fully outline how the graph will look and behave on any Cartesian plane.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so that they are easy to graph.
This form of a linear equation is particularly helpful because it gives you immediate information about the line.
The equation is expressed as:

• \( y = mx + b \)

where:

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

This format allows you to quickly identify how steep the line is and where it crosses the y-axis.

Understanding Slope

The slope, \( m \), tells you how fast or slow the line rises or falls.
For example, a slope of \( \frac{1}{3} \) means the line goes up 1 unit for every 3 units it goes right.
A positive slope indicates an upward trend from left to right, while a negative slope shows a downward trend.

Plotting Points

Plotting points involves placing them on a graph according to their coordinates.
To graph linear equations effectively, begin by plotting the y-intercept.
For the function \( h(x) = \frac{1}{3}x + 2 \), the y-intercept is 2, corresponding to the point \( (0, 2) \).

Using the Slope

After placing the y-intercept, use the slope to find another point.
If the slope is \( \frac{1}{3} \), you move 3 units horizontally to the right and 1 unit up.
This gives you the point \( (3, 3) \).
With these two points, you can draw a straight line through them.
Extend this line in both directions to show the linear function as it continues on the plane.
By understanding how to plot points using the slope-intercept form, you'll find graphing linear functions simple and intuitive.