Problem 28 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ g(x)=3 $$

Step-by-Step Solution

Answer

The slope is 0, and the y-intercept is 3.

Step 1: Identify the Type of Function

The given function is \( g(x) = 3 \), which is a constant function. This means the function does not depend on \( x \), and its graph will be a horizontal line.

Step 2: Determine the Y-Intercept

For the function \( g(x) = 3 \), the y-intercept is the value of \( g(x) \) when \( x = 0 \). Here, the y-intercept is 3. The graph crosses the y-axis at the point \( (0, 3) \).

Step 3: Identify the Slope

The slope of a constant function \( g(x) = c \) is 0 because the value of the function does not change as \( x \) changes. A horizontal line indicates no change in \( y \) over \( x \).

Step 4: Graph the Linear Function

To graph \( g(x) = 3 \), draw a horizontal line through the point \( (0, 3) \) on the y-axis. This line extends infinitely to the left and right for all values of \( x \).

Key Concepts

constant function

A constant function is one of the simplest types of linear functions. In a constant function, the output value does not change regardless of the input value. Mathematically, it is represented as \( g(x) = c \), where \( c \) is a constant number. In this scenario, the line created by the graph of the function is always horizontal. This is because no matter what value \( x \) takes, \( g(x) \) remains the same.

Here are a few key characteristics of constant functions:

• The graph is a straight, horizontal line.
• The slope of the graph is zero, indicating no rise or fall as \( x \) changes.
• There is only one point of intersection with the y-axis, which defines the y-intercept.

Understanding constant functions allows you to quickly determine the behavior of certain linear graphs and make predictions about outcome values.

y-intercept

The y-intercept is a critical aspect of understanding a linear function's graph.

In simpler terms, the y-intercept is where the graph of an equation crosses the y-axis. It is the value of the function when \( x \) is zero. For the function \( g(x) = 3 \), the y-intercept is 3, as the function remains constant at this value.

To visualize this:

• Look at the y-axis and find the point where the graph meets this vertical line.
• In our function, the graph crosses precisely at the point \((0, 3)\).
• This point gives a clear indication of where the function stands concerning the y-axis without requiring changes in \( x \).

Understanding the y-intercept is valuable because it acts as the starting point for graphing linear functions and gives insights into where a horizontal line stands on a graph.

slope

The slope of a line in a graph represents how steep or flat the line is. It's a measure of how much \( y \) changes with a change in \( x \).

For a constant function like \( g(x) = 3 \), the slope is extremely straightforward. It is zero. Here's why:

• A horizontal line implies there is no vertical increase or decrease, no matter the change in \( x \).
• The formula for slope, \( m = \frac{\Delta y}{\Delta x} \), means the change in \( y \) over the change in \( x \) is always zero, because \( \Delta y = 0 \).
• This results in a flat, horizontal line which makes the function predictable.

Understanding the concept of slope helps in determining the nature of the graph, showing whether a function is increasing, decreasing, or constant.