Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=\frac{1}{2} x$$

Step-by-Step Solution

Answer

The graph is a straight line passing through the origin with a slope of \(\frac{1}{2}\).

Step 1: Identify the Function Type

The function given is a linear function of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = \frac{1}{2} \) and \( b = 0 \). This means the graph of the function will be a straight line.

Step 2: Determine the Slope and Y-intercept

The slope of the function \( f(x) = \frac{1}{2}x \) is \( \frac{1}{2} \), which means for every 1 unit increase in \( x \), \( y \) increases by \( \frac{1}{2} \). The y-intercept is 0, which means the line passes through the origin (0, 0).

Step 3: Plot the Y-intercept

Plot the point (0, 0) on a coordinate plane. This point is where the line will intersect the y-axis.

Step 4: Use the Slope to Find Another Point

Starting from the y-intercept (0, 0), use the slope to find another point on the line. Since the slope is \( \frac{1}{2} \), move 1 unit to the right and \( \frac{1}{2} \) unit up to locate the next point, which is (1, \( \frac{1}{2} \)). Plot this point.

Step 5: Draw the Line

Draw a straight line through the points (0, 0) and (1, \( \frac{1}{2} \)). This line represents the graph of the function \( f(x) = \frac{1}{2}x \).

Key Concepts

Graphing Techniques

Graphing a linear function by hand can be made simple by following a few straightforward steps. First, start by identifying the type of function you have. In this case, we have a linear function. A linear function, as the name implies, will graph as a straight line on the coordinate plane. The next step is to gather information about the slope and y-intercept, which will guide how you sketch your graph. ### Steps for Graphing

• **Identify the points**: These help you position your line accurately. The key points you will need are the y-intercept and a point derived from the slope.
• **Draw using a ruler**: Straight lines should indeed be straight. Ensure that you connect your points properly.

By mastering these graphing techniques, you gain a powerful tool for visualizing mathematical relationships. This not only boosts your understanding, but also your ability to solve further related problems.

Slope-Intercept Form

The slope-intercept form is a popular form of expressing linear functions. It comes in the format of \( y = mx + b \), where **\( m \)** is the slope and **\( b \)** is the y-intercept. This form is invaluable because it immediately tells you two critical pieces of information: how steep the line is (the slope), and where it crosses the y-axis (the y-intercept). ### Slope (\( m \))

• **Definition**: The slope represents how much the line "rises" vertically for each unit it moves "across" horizontally.
• **Example**: For the function \( f(x) = \frac{1}{2}x \), the slope is \( \frac{1}{2} \). This means that for a movement of 1 unit horizontally, the line moves up by \( \frac{1}{2} \) units.

### Y-intercept (\( b \))

• **Definition**: The point where the line crosses the y-axis.
• **Example**: In \( f(x) = \frac{1}{2}x \), the y-intercept is 0, implying the line passes through the origin (0, 0).

Understanding the components of slope-intercept form can greatly simplify the process of sketching graphs and analyzing linear functions.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graphically represent equations. It is divided into four quadrants by the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is described by a pair of numbers \((x, y)\), known as coordinates. ### Features of the Coordinate Plane

• **Axes**: The x-axis runs horizontally, and the y-axis runs vertically. Their intersection point is known as the origin (0,0).
• **Quadrants**: The plane is divided into four quadrants: I, II, III, and IV, moving counterclockwise from the upper right.
• **Plotting Points**: Each point represented as \((x, y)\) can be accurately located on the plane by moving \(x\) units along the x-axis and \(y\) units along the y-axis.

When graphing linear functions, using the coordinate plane allows you to clearly visualize the relation between variables. With practice, it becomes intuitive to see how changes in the equation affect the graph's appearance.