Problem 69 - Algebra 2

Question

Graph each function \(f(x)=5\)

Step-by-Step Solution

Answer

The graph of \( f(x) = 5 \) is a horizontal line at \( y=5 \).

Step 1: Identify the type of function

The function given is \( f(x) = 5 \). This is a constant function, as the value of \( f(x) \) does not change regardless of the input \( x \).

Step 2: Understand the characteristics of the graph

For the function \( f(x) = 5 \), the graph will be a horizontal line. Every point on this line will have the form \( (x, 5) \) where \( x \) can be any real number.

Step 3: Note the y-intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. For \( f(x) = 5 \), the graph crosses the y-axis at the point (0, 5).

Step 4: Plot the graph

On a graph, draw a horizontal line passing through the y-axis at the point 5 on the y-axis. This line will extend infinitely in both horizontal directions, representing that \( f(x) = 5 \) for all x values.

Key Concepts

Horizontal Line

A horizontal line is a type of line that remains constant across its entire length. In the case of the graph of the function \( f(x) = 5 \), this line appears as a flat, unchanging line parallel to the x-axis. No matter how far you extend it on the graph, it never rises or falls. What makes a horizontal line special is that every point on the line has the same y-coordinate—in this case, 5.

• Each point on the line can be represented as \( (x, 5) \), where \( x \) is any real number.
• This uniformity means the slope of the line is 0, denoting no change in value as you move along the x-axis.
• Such lines are associated with constant functions, where the output value \( f(x) \) remains the same regardless of the input \( x \).

In summary, the graph of \( f(x) = 5 \) is a quintessential example of a horizontal line that perfectly embodies consistency.

Graphing Functions

When graphing functions, it's essential to understand the relation between the variables and how they translate onto the coordinate system. For a constant function like \( f(x) = 5 \), the graphing process is simplified because the function does not change as \( x \) varies.

• Start by identifying the nature of the function—in this case, it is a constant.
• Next, consider key points where the graph should appear. With constant functions, all these points will lie on a horizontal line.
• Plotting involves marking the y-coordinate, 5, across a range of x-values. You can choose any x-values since the corresponding y-values are always 5.
• Joining these points will yield the horizontal line representing the function.

Graphing functions becomes intuitive with practice, and understanding simple cases like constant functions lays the foundation for more complex scenarios.

Y-Intercept

The y-intercept is a crucial concept in graphing as it represents the point where a graph intersects the y-axis. To find the y-intercept for a function, you evaluate the function at \( x = 0 \).

• For the function \( f(x) = 5 \), substitute \( x = 0 \) to find that \( f(0) = 5 \).
• This tells us the y-intercept is the point (0, 5), where the horizontal line crosses the y-axis.
• The y-intercept serves as an anchor point from where the graph extends horizontally across the y-value of 5.

In essence, understanding the y-intercept helps you to quickly locate the starting point of the graph on the y-axis, making it easier to plot the rest of the function.