Problem 83 - College Algebra Essentials

Question

will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ f(x)=-2 $$

Step-by-Step Solution

Answer

The graph of the function \( f(x) = -2 \) is represented as a horizontal line that intersects the y-axis at the point where y equals to -2.

Step 1: Understand the Function

Firstly, understand the function provided, i.e., \( f(x) = -2 \). This function implies that, for any value of x, the value of f(x) will be -2. This indicates that our line will be horizontal and will intersect the y-axis at -2.

Step 2: Draw the coordinate Axes

Draw a horizontal x-axis and a vertical y-axis. Mark the point -2 on the y-axis as this is where our line will pass through.

Step 3: Draw the Line

Draw a horizontal line which passes through the point -2 in y-axis. This drawn line represents the function \( f(x) = -2 \) in graphical form.

Key Concepts

Coordinate Axes

Imagine a map that could represent every possible location for a treasure, with each spot marked by coordinates. In mathematics, we have a similar map for plotting functions, and this is called the coordinate axes. These axes consist of two perpendicular lines that intersect at a point called the origin.

The horizontal line is known as the x-axis and it measures the horizontal position, left or right, of a point. The vertical line is called the y-axis and it determines the vertical position, up or down, of a point. Whenever we want to graph a function like a linear function, we start by drawing these axes on graph paper. They are the fundamental building blocks for graphing functions because they provide a reference framework where each point is uniquely specified by its x (horizontal) and y (vertical) coordinates.

Horizontal Line Graph

Let's take the flat horizon you see when gazing out across the sea as an analogy for a horizontal line graph. In the same way the horizon has a constant elevation, a horizontal line graph has a constant y-coordinate. When we have a linear function like \( f(x) = -2 \), it means no matter how far you travel along the x-axis, the height, or y-coordinate, does not change. So, the graph is a straight horizontal line.

To sketch this, simply find the constant y-coordinate on the y-axis, which is -2 in this case, and draw a straight line through it that runs parallel to the x-axis. Remember, this line extends infinitely in both the left and right directions and every point on this line has the same y-coordinate, showing us visually that the function's value does not depend on \( x \).

Constant Function

Imagine you set a glass of water on a table and it stays at the same level all day. This uniform level is just like a constant function in mathematics – it doesn't change regardless of time or conditions. When we see a function like \( f(x) = -2 \), it's saying, 'No matter what x you give me, I'm always going to be -2.' This is the essence of a constant function: it produces a single, unchanging output for any input.

Graphically, this constancy is represented by a horizontal line. The function essentially flatlines, showing no rise or fall as x changes, which is why its graph doesn't have a slope. A constant function effectively tells a straightforward story: 'It doesn't matter where you are, the answer will always be the same.'