Question

Graph each function. Be sure to label three points on the graph. $$f(x)=3$$

Short answer

The graph of \(f(x) = 3\) is a horizontal line at \(y = 3\), with points \((-1, 3)\), \((0, 3)\), and \((1, 3)\).

Step-by-step solution

Identify the function type

The given function is a constant function because it is in the form of \(f(x) = c\) where \(c\) is a constant. In this case, \(c = 3\).

Determine the y-value

For a constant function \(f(x) = 3\), every value of \(x\) will result in the same value of \(y\), which is 3.

Choose three points

Choose three values of \(x\) to plot. For simplicity, use \(x = -1, 0,\) and \(1\). Calculating, we get the points: - \((-1, 3)\)- \((0, 3)\)- \((1, 3)\)

Plot the points on a graph

On the Cartesian plane, plot the points \((-1, 3)\), \((0, 3)\), and \((1, 3)\). These points should all lie on the horizontal line corresponding to \(y = 3\).

Draw the graph

Draw a horizontal line connecting the points you have plotted. The line should extend in both directions horizontally, parallel to the x-axis, labeled at \(y = 3\).

Key concepts

constant function

A constant function is one of the simplest types of functions you will encounter in mathematics.
It is in the form of \(f(x) = c\), where \(c\) is a constant.
This means that no matter what value of \(x\) you use, the value of \(f(x)\) is always the same.

For example, in the exercise, we have \(f(x) = 3\).
Here, \(f(x)\) equals 3 for any value of \(x\).
So, whether \(x = -1, x = 0,\) or \(x = 5\), \(f(x)\) remains 3.
This results in a horizontal line on the graph.

It's crucial to understand that a constant function is different from other functions where the value of \(f(x)\) changes with \(x\).
The constant value makes the function easy to graph and analyze.

Constant functions are incredibly useful as they represent fixed values in real-world scenarios.

graphing functions

Graphing functions is a fundamental skill in algebra and pre-calculus.
It helps visualize the relationship between \(x\) and \(y\), or in this case, \(x\) and \(f(x)\).

To graph a function, follow these steps:

• Identify the type of function: Is it linear, quadratic, or constant?
• Determine key values: For constant functions, the key value is the constant itself.
• Select useful \(x\)-values: Pick points that are easy to compute.
• Plot the points: Place them on a Cartesian plane.
• Draw the graph: This could be a line, parabola, or other shapes depending on the function type.

In our example, plot points for \(f(x) = 3\) by choosing \(x = -1, 0,\) and \(1\).
Then, plot the coordinates \((-1, 3)\), \( (0, 3)\), and \( (1, 3)\) on the graph.
The points line up horizontally because the function is constant.

coordinate points

Coordinate points are essential when plotting graphs.
Each point has an \(x\)-value and a \(y\)-value.
For example, the point \((-1, 3)\) means \(x = -1\) and \(y = 3\).

When dealing with functions:

• The \(x\)-value is your input.
• The \(y\)-value is your output.

For the function \(f(x) = 3\), your \(y\)-value will always be the constant, \(3\), regardless of the \(x\)-value.

Here’s a straightforward way to choose and plot points:

• Pick a few \(x\)-values (e.g., \(-1, 0, 1\)).
• Calculate their corresponding \(y\)-values (which are all 3 in this case).
• Plot the points on the coordinate plane.
• Draw a line through the points.

In our example: Plotting \((-1, 3)\), \( (0, 3)\), and \( (1, 3)\) helps visualize the constant function.