Problem 16 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. \(y=-1.5\)

Step-by-Step Solution

Answer

The function is a linear function, graphed as a horizontal line at \(y = -1.5\).

Step 1: Identify the Type of Function

The given equation is \(y = -1.5\). This is in the form of \(y = c\), where \(c\) is a constant. Equations of the form \(y = c\) are linear functions that represent horizontal lines on the graph.

Step 2: Graph the Function

To graph the function \(y = -1.5\), draw a horizontal line at \(y = -1.5\) on the Cartesian plane. Every point on this line will have the same y-coordinate, \(-1.5\), while the x-coordinate can be any real number.

Key Concepts

Horizontal Line

A horizontal line is a straight line that runs from left to right across the Cartesian coordinate plane. One key characteristic of a horizontal line is that it has a constant y-value for every point along the line.
This means that although you can pick any x-value, the y-value remains unchanged based on the equation of the line.In our given problem, the equation is expressed as \( y = -1.5 \). This indicates that no matter where you move along the horizontal axis (the x-axis), the line is always positioned 1.5 units below the origin on the vertical axis.Horizontal lines are unique because:

• They have a slope of zero, meaning they do not rise or fall as they extend in either direction.
• They represent levels of constancy or uniformity within a dataset or function.

In practical terms, consider how a line of this nature can illustrate a fixed attribute, such as maintaining a constant temperature or pressure over time.

Constant Function

A constant function is a type of linear function where the output value remains the same regardless of the input. The general form of a constant function can be articulated as \( y = c \), where \( c \) is a fixed number.In our example, \( y = -1.5 \) is a perfect representation of a constant function because, for any x-value you choose, the output will constantly be -1.5.Here are some essential traits of constant functions:

• They graph as horizontal lines on a coordinate plane.
• The derivative of a constant function is zero since there is no change in value.
• Constant functions are useful in scenarios where one needs to define a uniform behavior across all inputs.

So, in a mathematical or real-world context, constant functions depict scenarios with no variation, which can simplify analysis and interpretation of data.

Graphing Equations

Graphing equations is a fundamental aspect of understanding and visualizing functions. It involves plotting points on a coordinate plane to represent the solutions of the equation.For linear functions like a constant function, the process is straightforward. For \( y = -1.5 \), you draw a horizontal line that crosses the y-axis at -1.5. Here’s a simple guide:

• Choose several x-values – they can be any real numbers since the choice of x doesn’t affect the y-value.
• Plot the points on the graph with your chosen x-values paired with the constant y-value of -1.5.
• Connect the plotted points with a straight line extending in both directions to complete the graph.

The horizontal nature of the line indicates that the function doesn't change no matter how x varies. Graphing is especially helpful in disciplines like physics and engineering, where it's crucial to interpret and predict trends and behaviors visually.