Problem 21 - A Graphical Approach to College Algebra

Question

Sketch the graph of \(f\) by hand. $$f(x)=\frac{1}{2} x$$

Step-by-Step Solution

Answer

Sketch a line through (0,0) and (2,1). Thus, complete the graph.

Step 1: Identify the Function Type

The function given is in the form \( f(x) = mx + b \) which represents a linear function. In this case, \( m = \frac{1}{2} \) is the slope, and \( b = 0 \) is the y-intercept.

Step 2: Determine the Y-Intercept

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation: \( f(0) = \frac{1}{2} \times 0 = 0 \). Thus, the y-intercept is the point \((0, 0)\).

Step 3: Determine the Slope

The slope of the line, \( m = \frac{1}{2} \), means that for every 1 unit increase in \( x \), \( f(x) \) increases by \( \frac{1}{2} \) unit. This helps determine another point on the graph.

Step 4: Find Another Point Using the Slope

From the y-intercept at \((0, 0)\), apply the slope. Increasing \( x \) by 2 produces \( f(2) = \frac{1}{2} \times 2 = 1 \). Thus, another point on the graph is \((2, 1)\).

Step 5: Sketch the Graph

Plot the points \((0, 0)\) and \((2, 1)\) on the coordinate plane. Draw a straight line through these points extending in both directions. This line represents the graph of \( f(x) = \frac{1}{2} x \).

Key Concepts

Linear Function

A linear function is a type of function where each variable is multiplied by a constant and possibly combined with a constant term. It's represented in the form \( f(x) = mx + b \), where:

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

Linear functions graph as straight lines. They do not have any exponents other than 1, and they don't curve or angle unexpectedly. These functions are useful because they are predictable and show a constant rate of change. In our exercise, the function \( f(x) = \frac{1}{2} x \) is a classic linear function. This means the graph will be a straight line, indicating that any change in \( x \) results in a proportional change in \( f(x) \). Understanding linear functions lays the foundation for exploring more complex types of functions.

Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). This format neatly highlights important features:

• \( m \) is the slope, indicating the steepness of the line
• \( b \) is the y-intercept, where the line crosses the y-axis

This form is particularly useful because it shows both the starting point and the rate of change at a glance. For example, the line's equation \( f(x) = \frac{1}{2} x \) tells us the slope \( m = \frac{1}{2} \). This means as \( x \) increases by 1, \( f(x) \) increases by \( \frac{1}{2} \). The absence of a constant term \( b \) means the line passes through the origin.

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can plot points to represent algebraic relationships. It consists of an x-axis (horizontal) and a y-axis (vertical), which intersect at the origin point (0,0).
To sketch a graph using a function like \( f(x) = \frac{1}{2} x \), you plot points that satisfy the function and then draw a line through them.
In this case, important points include:

• The y-intercept \((0,0)\)
• Another point, calculated using the slope, such as \((2,1)\)

Once these points are plotted, you can draw a straight line to form the graph of the function.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the equation \( y = mx + b \), the \( b \) represents the y-intercept. It is found by setting \( x = 0 \) in the equation.
In our function \( f(x) = \frac{1}{2} x \), substituting \( x = 0 \) gives us \( f(0) = 0 \). Thus, the y-intercept occurs at the coordinate \((0, 0)\).
The y-intercept is crucial for sketching a graph because it provides a starting point. Understanding how the y-intercept works aids in quickly visualizing how linear functions behave on the coordinate plane.