Problem 72 - Algebra 2

Question

Identify each equation as a type of function. Then graph the equation. $$ y=8 $$

Step-by-Step Solution

Answer

The equation is a horizontal line and linear function. Its graph is a line at \( y = 8 \).

Step 1: Identify the Function Type

The given equation is \( y = 8 \). This equation represents a horizontal line, which is a specific case of a linear function. In a linear function of the form \( y = mx + b \), the slope \( m \) is 0, and the y-intercept \( b \) is 8.

Step 2: Graph the Equation

To graph the equation \( y = 8 \), draw a horizontal line across the Cartesian plane that crosses the y-axis at the point (0, 8). Every point on this line has the y-coordinate 8, indicating that it is parallel to the x-axis.

Key Concepts

Graphing Functions

Graphing functions is a visual way to represent the relationship between variables in an equation. When you have a function like \( y = f(x) \), you can plot points on a graph that show how \( y \) changes with \( x \). This allows you to see patterns, trends, and the overall behavior of the function.
Here’s how to graph a function:

• Choose values for \( x \) and compute the corresponding \( y \) values using the function's equation.
• Plot these \( (x, y) \) coordinates on the graph.
• Connect the points to see the shape of the graph, which helps in understanding the function's nature.

For the equation \( y = 8 \), since \( y \) does not depend on \( x \), every value of \( x \) will give the same \( y \)-value. This leads to a horizontal line on the graph, which we’ll explore further below. Graphing functions is a helpful tool for visual learners and gives insight into the equation's effects.

Horizontal Line

A horizontal line is a straight line that goes from left to right across a graph. In the equation \( y = 8 \), the line is horizontal because \( y \) is constant, meaning it does not change no matter what \( x \) is.
Here are some important things to remember about horizontal lines:

• The slope of a horizontal line is 0 because there is no vertical change as \( x \) changes.
• All points on the line have the same \( y \)-coordinate, which in this case is 8.
• Horizontal lines are parallel to the x-axis.

When graphing \( y = 8 \), you simply draw a line parallel to the x-axis that crosses the y-axis at (0,8). This type of line is a great example of a simplified linear function where the slope is zero. It shows how certain linear functions can be extremely simple, yet conceptually important for understanding relationships between variables.

Cartesian Plane

The Cartesian Plane is a two-dimensional space defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is identified by its coordinates \((x, y)\), which tell you how far along and how far up or down the point is from the origin (0,0).
Here’s why understanding the Cartesian Plane is essential:

• It serves as the foundation for graphing functions and analyzing their properties.
• In the plane, the x-axis and y-axis divide the space into four quadrants where points can be plotted.
• Linear functions, like horizontal lines, can easily be represented by plotting points and connecting them.

When you graph the equation \( y = 8 \) on the Cartesian Plane, you focus on the y-coordinate being consistently 8. This approach makes it easy to visualize and understand how horizontal lines and other functions behave within this grid system. The Cartesian Plane is a versatile tool for seeing how different mathematical concepts come to life visually.