Problem 70 - Intermediate Algebra

Question

Graph each of the basic functions. $$ S(x)=x $$

Step-by-Step Solution

Answer

The graph of \( S(x) = x \) is a straight line passing through the origin with a slope of 1.

Step 1: Understand the Basic Function

The given function is the identity function, denoted as \( S(x) = x \). This is one of the simplest linear functions where the output value is directly equal to the input value. Its graph is a straight line passing through the origin.

Step 2: Identify Key Points

For linear functions like \( S(x) = x \), any two points are sufficient to draw the graph. Commonly used points for this function are \((0, 0)\) and \((1, 1)\), since substituting \( x = 0 \) gives \( S(0) = 0 \), and substituting \( x = 1 \) gives \( S(1) = 1 \).

Step 3: Plot the Points on a Coordinate Plane

Mark the points \((0, 0)\) and \((1, 1)\) on the Cartesian coordinate plane. Point \((0, 0)\) is the origin, and \((1, 1)\) is one unit to the right and one unit up from the origin.

Step 4: Draw the Line

Use a ruler to draw a straight line through the marked points \((0, 0)\) and \((1, 1)\). This line represents the graph of the function \( S(x) = x \) and continues infinitely in both directions.

Step 5: Verify the Line's Slope and Direction

The slope of the line \( S(x) = x \) is 1, indicating a 45-degree angle with the x-axis. The line should have a positive slope, going upward as it moves from left to right, confirming that any further points would lie on this line (e.g., \((2, 2)\)).

Key Concepts

Identity Function

An identity function is one of the simplest forms of functions in mathematics, represented by the equation \( S(x) = x \). In this function, each input value \( x \) is directly mapped to itself, making the output \( y \) equal to the input. This means that if you input 5, the output is also 5, and so on.

• This function is linear because it can be graphically represented by a straight line.
• The identity function is characterized by having a slope of 1.
• The line passes through the origin \((0, 0)\), because when \( x = 0 \), \( S(0) = 0 \).

A useful feature of the identity function is that it serves as a benchmark to compare other functions. Its simplicity makes it an excellent starting point in understanding linear functions, as any deviation from the line \( y = x \) in other functions illustrates a transformation such as a shift, stretch, or compression.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is your canvas for graphing any function, including the identity function. It consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, which is located at the point \((0, 0)\).

• Each point on the coordinate plane is identified by an ordered pair \((x, y)\).
• As you move right from the origin on the x-axis, the values of \( x \) increase, and as you move left, they decrease.
• On the y-axis, values increase as you move up and decrease as you move down.

When graphing a linear function like the identity function, you plot points on this plane and connect them. In the case of the identity function, such points might include \((0,0)\), \((1,1)\), \((2,2)\), and so forth, forming a straight line that perfectly bisects the first quadrant, continuing infinitely in both directions.

Linear Equations

Linear equations are equations that graph as straight lines on the coordinate plane. A simple example of a linear equation is the identity function \( S(x) = x \).

• These equations can typically be written in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• For the identity function, \( m = 1 \) and \( c = 0 \), meaning the line crosses the origin and rises at a 45-degree angle.
• The slope \( m \) tells you how steep the line is, with positive values indicating an upward tilt and negative values a downward tilt.

Understanding linear equations is crucial, as they are foundational in higher-level math subjects and have real-world applications in fields such as economics, physics, and engineering. Mastery of graphing these functions can simplify complex data sets and provide a visual understanding of relationships between variables.