Problem 87 - Algebra and Trigonometry

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 a horizontal line passing through the y-coordinate -2.

Step 1: Understanding the Function

The function given is \( f(x)=-2 \). Here, the value of \( f(x) \), also known as the 'y' value, is always -2, regardless of the 'x' value.

Step 2: Plotting the Points

A few points that lie on this function are (-2, -2), (0, -2), and (2, -2), among others. All these points have a 'y' value of -2.

Step 3: Drawing the Graph

By connecting the points plotted in step 2, on the Cartesian plane, a horizontal line parallel to the x-axis and passing through the y-coordinate -2 is formed. This line represents the graph of the function \( f(x) = -2 \).

Key Concepts

Graphing

Graphing is about visually representing mathematical functions on paper or digital tools. Think of it as drawing a picture to see how a function behaves. Graphing gives us a clear understanding of what mathematical expressions look like.

To start graphing a linear function like \( f(x) = -2 \), you begin with plotting key points from the function. Since this is a simple constant function, all of the points have the same y-value, -2, no matter what x-value you choose. This makes it very easy to draw.

• Choose multiple x-values; any numbers will work, such as -3, 0, and 3.
• For each x-value, the y-value remains -2, giving the points (-3, -2), (0, -2), and (3, -2).
• Mark these points on the graph.
• Connect the points in a straight line.

Remember, graphing not only helps you solve equations but also shows you the bigger picture of how these equations act. It's like watching the behavior of a function play out right before your eyes.

Horizontal Line

A horizontal line is one of the simplest types of graphs you can come across. It's a straight, flat line that runs from left to right across a graph.

For the function \( f(x) = -2 \), the graph is a perfect example of a horizontal line. Here’s what you need to remember:

• A horizontal line always runs parallel to the x-axis, not touching the x-axis unless the y-value is zero (e.g., \( y=0 \)).
• With a constant y-value, this line looks the same regardless of how far you extend it along the x-axis.
• In the graph of \( f(x) = -2 \), every point on the line has a y-coordinate of -2.

Horizontal lines are simple yet powerful. They instantly show that the y-value never changes, highlighting that there is no relationship between x and y in the linear sense.

Cartesian Plane

The Cartesian Plane is essentially the backdrop where all graphing takes place. It consists of two number lines that intersect at right angles.

Think of the Cartesian Plane as a grid with two axes:

• The x-axis, which runs horizontally (left to right).
• The y-axis, which runs vertically (up and down).

The point where they meet is called the origin, labeled as (0, 0).

For our exercise, the function \( f(x) = -2 \) is plotted on this plane. The Cartesian Plane allows us to see the relationship between x-values and y-values. When you plot the points (-2, -2), (0, -2), and (2, -2), you observe their alignment along the same y-coordinate, clearly visible on the grid.

Using a Cartesian Plane helps in understanding complex relationships between variables as well as visualizing simple ones like the horizontal line of \( f(x) = -2 \). It's indispensable for classifying points, observing intersections, and understanding how different lines interact.