Problem 16 - Intermediate Algebra

Question

Graph each of the following linear and quadratic functions. $$f(x)=1$$

Step-by-Step Solution

Answer

The graph of \( f(x) = 1 \) is a horizontal line at \( y=1 \).

Step 1: Understand the function type

The function given is \( f(x) = 1 \), which is a constant function. This is not a linear or quadratic function, but rather a horizontal line where the output value is the same for every \( x \).

Step 2: Identify key points

Since \( f(x) = 1 \) is constant, for any \( x \) value, \( f(x) \) will always be 1. A few key points can be \((-2, 1)\), \( (0, 1) \), and \( (2, 1) \).

Step 3: Draw the graph

On a coordinate plane, mark the points \((-2, 1)\), \((0, 1)\), and \((2, 1)\). Connect these points with a straight horizontal line across the graph. This line will extend infinitely in both the positive and negative directions of the x-axis.

Key Concepts

Constant Function

A constant function is one where the output value remains unchanged, regardless of the input. In mathematical terms, it's often expressed as \( f(x) = c \), where \( c \) is a fixed number. This means no matter what value you substitute in for \( x \), the result is always \( c \).

Graphically, a constant function appears as a horizontal line on the coordinate plane. No matter how far you move along the x-axis, the function’s value on the y-axis stays the same. For instance, if you have \( f(x) = 1 \), this means:

• The output is always 1 for any input.
• The graph line is parallel to the x-axis.

One key characteristic of constant functions is that their slope is zero. In the world of functions, the slope indicates how steeply a line rises or falls. Since constant functions don’t rise or fall, their slope is zero.

Linear Function

Linear functions are very straightforward and simple to understand. They describe a relationship where each input corresponds to a direct, proportionate change in output. A typical form for a linear function is \( f(x) = mx + b \).

In this equation:

• \( m \) represents the slope - this tells us how steep the line is.
• \( b \) represents the y-intercept - this is where the line crosses the y-axis.

If you picture a straight inclined or declined line across your graph, you’re visualizing a typical linear function. This line can rise, fall, or even be perfectly horizontal, reflecting the function’s slope. It's essential to remember that the linear slope determines whether a line climbs upwards or slopes downwards as you move from left to right.

Coordinate Plane

The coordinate plane is a two-dimensional surface that helps us visually represent mathematical functions using two perpendicular lines, typically called the x-axis and the y-axis. When graphing functions like the constant or linear functions, this is where we effectively plot their behavior.

Key components of the coordinate plane include:

• The x-axis, which is horizontal, represents all possible input values.
• The y-axis, which is vertical, represents the outputs.
• The origin, the point (0, 0), the intersection of the x-axis and y-axis.

The coordinate plane allows for a broader understanding of equations as geometric shapes. For example, given the function \( f(x) = 1 \), drawing points at various x-values, all resulting in the same y-value (1), easily shows the function’s constancy as a horizontal line.