Question

Sketch the graph of the function \(\mathrm{H}=\\{(\mathrm{x}, \mathrm{H}(\mathrm{x})) \mid \mathrm{H}(\mathrm{x})=6\\}\).

Short answer

The graph of the function \(H(x) = 6\) is a horizontal straight line parallel to the x-axis, passing through the point (0, 6), and having the equation \(H(x) = 6\) for all values of x.

Step-by-step solution

Understand the function

The given function is H(x) = 6, which means that the value of H(x) remains constant at 6 for all values of x.

Set up the graph

To sketch the graph, first draw the axes with labeled x and H(x) (or y) axis. Choose an appropriate scale.

Plot a few points

Choose a few values of x and calculate the corresponding values of H(x) using the given equation H(x) = 6. For example: - For x = -3, H(x) = 6. - For x = 0, H(x) = 6. - For x = 3, H(x) = 6.

Draw the graph

Plot the points from Step 3 on the graph and connect them with a straight horizontal line. You will see that the graph is a horizontal straight line parallel to the x-axis, passing through the point (0, 6) and having the equation H(x) = 6 for all values of x.

Key concepts

Horizontal Line

A horizontal line in a graph represents a constant function, meaning its value does not change regardless of the input. In the expression given, the function \(H(x) = 6\), the value of \(H(x)\) is always 6 no matter what value \(x\) takes. In simpler terms, no matter where you are on the x-axis, the output (or the y-value) will always be 6.

This results in a line that is parallel to the x-axis since there is no ascent or descent as x varies.
Thus, as you move from left to right across the graph, the line stays perfectly level, showcasing that the function’s value is unchanging.

Function Notation

Function notation allows us to express outputs based on inputs succinctly. For example, in \(H(x) = 6\), the \(H\) refers to the function name, and \(x\) is known as the independent variable or input. The notation \(H(x)\) is read as "H of x," which indicates the output of function \(H\) for a given input \(x\).

Using this format helps in easily substituting values into the function to find corresponding outputs.
Here, regardless of what \(x\) is—such as -3, 0, or 3—the outcome remains constant at 6, indicating every input leads to the same output, further reinforcing the concept of a horizontal line on the graph.

Graph Sketching

Graph sketching is the art of drawing a rough graph of a function based on several known characteristics.
Begin by understanding the type of function. Here, a constant function like \(H(x) = 6\), will always be horizontal on a Cartesian plane. As you sketch, start by marking the axes clearly to have a reference frame.

After drawing axes, select a few values of \(x\) and find their corresponding \(H(x)\).
For instance:

• If \(x = -3\), then \(H(x) = 6\).
• For \(x = 0\), it's \(H(x) = 6\).
• With \(x = 3\), it’s still \(H(x) = 6\).

Plot these points, and you'll observe they form a horizontal line across the same y-value. Connecting these plotted points provides a visual confirmation of the function’s behavior.

Axes Labeling

Axes labeling is crucial in graph representation to avoid confusion and convey correct information.
On a graph, the x-axis typically represents the independent variable, while the y-axis represents the dependent variable or output, such as \(H(x)\).

Labeling should be done clearly, with each axis indicating the variable it represents.
For this specific function \(H(x) = 6\), the x-axis might have tick marks for values like -3, 0, 3, and the y-axis should repeatedly show the value 6 for these x-values.
Moreover, having a marked point like (0, 6) aids in easily understanding the intersection of the graph and axes, making interpretation and communication straightforward.