Problem 55 - Precalculus Essentials

Question

Graph equation in a rectangular coordinate system. $$f(x)=1$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 1\) is a horizontal line crossing the y-axis at (0,1). For all x-values, the y-value is equal to 1.

Step 1: Identify type of function

Recognize that \(f(x) = 1\) is a constant function. In a constant function, each x-value in the domain corresponds to the same y-value.

Step 2: Identify the y-intercept

Since it's a constant function and the equation is \(f(x) = 1\), therefore, the line crosses the y-axis at (0,1). This point is called the y-intercept of the line.

Step 3: Sketch the horizontal line

On a set of axes, draw a horizontal line across the graph at \(y = 1\). This line includes all points whose y-coordinate is 1, and is continuous because for every x-coordinate, the y-coordinate is 1.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by an x-axis and y-axis. These axes are perpendicular to each other, where the x-axis runs horizontally and the y-axis runs vertically. Every point within this system can be specified by an ordered pair of numbers (x, y), which represent its coordinates. The point where both axes intersect is called the origin, labeled as (0, 0).

To plot any function or equation in this system, one needs to understand how to read and locate points based on their x and y coordinates. For example, the coordinates (3, 2) mean that we move 3 units along the x-axis (to the right if positive, to the left if negative) and then 2 units up along the y-axis (down if the number is negative). Visualization and accuracy in this system are key steps in understanding and drawing graphs for various mathematical functions.

Constant Function

A constant function is a type of mathematical function where the output value is the same, no matter what input value (x) you have. For any value of x, the function gives back a single, constant value. In algebra, constant functions are represented by equations such as f(x) = c, where c is a constant number. The graph of a constant function is particularly simple – it's a horizontal line on the rectangular coordinate system.

Understanding Function Behavior

Unlike other types of functions that can have peaks, valleys, slopes, and curves, constant functions are stable. They don't change as x changes, which is a significant property when analyzing function behavior. Constant functions are helpful in real-world contexts where consistent outcomes or fixed relations are discussed.

Y-Intercept

The y-intercept is a fundamental concept in algebra and coordinates geometry. It represents the point where a line or curve crosses the y-axis of the rectangular coordinate system. This point details where the graph of the function has an input value of x equal to zero. For the equation of a line in slope-intercept form, y = mx + b, the y-intercept is given by the value of b.

When it comes to constant functions, identifying the y-intercept is straightforward since the entire line lies parallel to the x-axis. For the constant function f(x) = c, the y-intercept is always (0, c). It's crucial to pinpoint the y-intercept as it often serves as a starting point in graphing the linear equation or can be a significant feature in understanding the function's behavior in context.

Horizontal Line Graph

Graphing a constant function results in a horizontal line graph. This type of graph is a visual representation of all the points (x, y) where the y-coordinate remains the same no matter the x-coordinate. For the function f(x) = 1, the graph will be a horizontal line at the height where y equals 1, extending infinitely to the left and right along the line y = 1.

It's important not to confuse the horizontal line graph of a constant function with other types of lines that can slope upwards or downwards. A horizontal line demonstrates no slope; in mathematical terms, its slope is 0. Knowing how to draw a horizontal line graph is fundamental because it helps in instantly recognizing constant functions, and it can represent steady states or equilibrium situations in real-world scenarios.