Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=3$$

Step-by-Step Solution

Answer

The graph is a horizontal line at \(y = 3\).

Step 1: Understanding the Function

The function given is a constant function, meaning that no matter what value of \(x\) you choose, \(f(x)\) will always be equal to the same constant value. Here, \(f(x) = 3\), which means for any \(x\), \(f(x)\) is 3.

Step 2: Identify Key Characteristics

In this function, there are specific characteristics to note: the graph will be a horizontal line because it is a constant function, the line does not have any intercepts aside from the y-intercept, which is 3, and there is no slope.

Step 3: Sketch the Horizontal Line

Plot the point (0,3) on the graph to mark the y-intercept. Because the function is constant, simply draw a straight horizontal line through this point parallel to the x-axis. The entire line is the graph of \(f(x) = 3\).

Step 4: Labeling and Final Touches

Label the horizontal line with \(f(x) = 3\) to indicate the function. Additionally, ensure that the axes are labeled with \(x\) and \(y\) to provide context for the graph.

Key Concepts

Graph Sketching

Graph sketching is a crucial skill in understanding the visual representation of mathematical functions. It provides a clear picture of how a function behaves for different inputs. When tasked with sketching a graph by hand, one must identify the type of function they are dealing with. For instance, the function \( f(x) = 3 \) is a constant function.

• A constant function, by definition, returns the same value, regardless of the input \( x \).
• The graph is typically characterized as a horizontal line.
• It simplifies the sketching process because you do not need to calculate different points since the output remains unchanged.

To sketch the graph of \( f(x) = 3 \), begin by identifying its form as a horizontal line. Next, locate the point where the line crosses the y-axis at \( (0, 3) \). Draw the line parallel to the x-axis because the function value doesn't change for different \( x \) values.
Graph sketching is more freehand and visual but requires an understanding of function properties to get it right.

Horizontal Line

A horizontal line on a graph is a simple yet powerful representation. It signifies that a function has a constant output value, no matter what the input variable is. In the case of \( f(x) = 3 \), the function defines a horizontal line.

• The horizontal nature comes from the absence of any variables affecting the function value – it remains constant.
• This line will run parallel to the x-axis, showing uniformity across any potential \( x \) values.
• In terms of slope, a horizontal line has a slope of zero as the rise is 0 over any run.

Visualizing this is straightforward: you only need to set your line to intercept the y-axis at 3 and then draw straight along horizontally. No matter how far you extend this line, its value remains 3.

Y-Intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. It is a key feature to understand as it provides a starting point for drawing the function.
For the constant function \( f(x) = 3 \), the y-intercept is easy to identify because

• This occurs at the point (0,3) since this is where \( x = 0 \) and \( f(x) \) equals 3.
• It denotes the sole intersection point with the y-axis, essential for drawing the horizontal line effectively.
• Once you plot this, you can extend the line horizontally from this point.

The y-intercept acts as a pivotal anchor for graph sketching, especially when dealing with constant functions. Understanding where a graph meets the y-axis simplifies the sketching process for many functions.