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Chapter 2: Problem 4

Match each linear equation with its graph $$ f(x)=2 $$

Short Answer

Expert verified
The graph of \( f(x) = 2 \) is a horizontal line at \( y = 2 \).

Step by step solution

01

Understanding the Equation

The given equation is a constant function, \( f(x) = 2 \). In this type of equation, the output is always constant regardless of the input \( x \). This means that no matter what value of \( x \) you choose, \( f(x) \) will always equal 2.
02

Analyzing the Graph Type

Since \( f(x) = 2 \) is a constant function, its graph is a horizontal line. A horizontal line means that the value of \( y \), which in this case is \( f(x) \), does not change and remains constant at 2.
03

Drawing the Graph

Locate the point where \( y = 2 \) on the y-axis. From this point, draw a straight horizontal line parallel to the x-axis. This line represents all the points where the output of the function is equal to 2 for any input \( x \).
04

Matching with the Graph

On the graph options provided, identify the graph that is a horizontal line crossing the y-axis at 2. This graph should run parallel to the x-axis, indicating that \( f(x) = 2 \) for all values of \( x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Function
A constant function is one of the simplest types of functions you will encounter. In a constant function, expressed by the equation \( f(x) = c \), the "c" represents a constant value that the function always outputs. No matter what input value you select for \( x \), the function will always yield the same constant output. In the context of our problem, this constant is 2, as seen in \( f(x) = 2 \).
  • This function does not depend on the input \( x \).
  • The output is a single, unchanging value.
  • Representing this graphically results in a line where \( y \) (or \( f(x) \)) is constant.
This makes constant functions unique, as any change or transformation of \( x \) has no impact on the overall output of the function.
Horizontal Line Graph
The graph of a constant function is a horizontal line. When you draw the function \( f(x) = 2 \), you will plot a series of points where the output (or \( y \)-value) remains at 2, forming a straight line parallel to the x-axis. Here, the entire line lies at \( y = 2 \), showing that no matter the \( x \)-coordinate, the value of \( y \) does not change.
  • Horizontal lines represent functions with no variation in output.
  • They run left to right across the graph, remaining equidistant from the x-axis.
  • For \( f(x) = 2 \), the line crosses the y-axis at 2, illustrating the constant output.
This horizontal line is an essential image of how constant functions appear visually, showing the function's unchanging nature.
Output Value Analysis
Analyzing the output values of a function helps in understanding its behavior. For the constant function \( f(x) = 2 \), the output is always 2, regardless of what \( x \) value you substitute into the function. This makes analysis straightforward, as all inputs lead to the same predetermined outcome.
  • The output tells us about the 'effect' of the function.
  • In constant functions, this effect doesn't vary, highlighting their predictability.
  • Each point on the graph mirrors this consistent result.
Understanding output value analysis for constant functions reinforces the idea that these functions reflect stability and lack the variability seen in other functions.

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